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The transversal number Τ (H) of a hypergraph H is the minimum number of vertices that intersect every edge of H. A 6-uniform hypergraph has all edges of size 6. On 10 November 2000 Tuza and Vestergaard [Discuss. Math. Graph Theory, 22 (2002), pp. 199-210] conjectured that if H is a 3-regular 6-uniform hypergraph of order n, then Τ(H) < 1/4n. In this paper we prove this conjecture, which has become known as the Tuza-Vestergaard conjecture. [ABSTRACT FROM AUTHOR]

A queue layout of a graph consists of a linear order of its vertices and a partition of its edges into queues, so that no two independent edges of the same queue are nested. The queue number of a graph is the minimum number of queues required by any of its queue layouts. A long-standing conjecture by Heath, Leighton and Rosenberg [SIAM J. Discrete Math., 5 (1992), pp. 398--412] states that the queue number of planar graphs is bounded. This conjecture has been partially settled in the positive for several subfamilies of planar graphs (most of which have bounded treewidth). In this paper, we make a further important step towards settling this conjecture. We prove that planar graphs of bounded degree (which may have unbounded treewidth) have bounded queue number. A notable implication of this result is that every planar graph of bounded degree admits a three-dimensional straight-line grid drawing in linear volume. Further implications are that every planar graph of bounded degree has bounded track number, and that every k-planar graph (i.e., every graph that can be drawn in the plane with at most k crossings per edge) of bounded degree has bounded queue number. [ABSTRACT FROM AUTHOR]

A variety of powerful extremal results have been shown for the chromatic number of triangle-free graphs. Three noteworthy bounds are in terms of the number of vertices, edges, and maximum degree given by Poljak and Tuza [SIAM J. Discrete Math., 7 (1994), pp. 307--313] and Johansson. There have been comparatively fewer works extending these types of bounds to graphs with a small number of triangles. One noteworthy exception is a result of Alon, Krivelevich, and Sudakov [J. Combin. Theory Ser. B, 77 (1999), pp. 73--82] bounding the chromatic number for graphs with low degree and few triangles per vertex; this bound is nearly the same as for triangle-free graphs. This type of parametrization is much less rigid and has appeared in dozens of combinatorial constructions. In this paper, we show a similar type of result for X (G) as a function of the number of vertices n, the number of edges m, as well as the triangle count (both local and global measures). Our results smoothly interpolate between the generic bounds true for all graphs and bounds for triangle-free graphs. Our results are tight for most of these cases; we show how an open problem regarding fractional chromatic number and degeneracy in triangle-free graphs can resolve the small remaining gap in our bounds. [ABSTRACT FROM AUTHOR]

We present support theory, a set of techniques for bounding extreme eigenvalues and condition numbers for matrix pencils. Our intended application of support theory is to enable proving condition number bounds for preconditioners for symmetric, positive definite systems. One key feature sets our approach apart from most other works: We use support numbers instead of generalized eigenvalues. Although closely related, we believe support numbers are more convenient to work with algebraically. This paper provides the theoretical foundation of support theory and describes a set of analytical tools and techniques. For example, we present a new theorem for bounding support numbers (generalized eigenvalues) where the matrices have a known factorization (not necessarily square or triangular). This result generalizes earlier results based on graph theory. We demonstrate the utility of this approach by a simple example: block Jacobi preconditioning on a model problem. Also, our analysis of a new class of preconditioners, maximum-weight basis preconditioners, in [E. G. Boman, D. Chen, B. Hendrickson, and S. Toledo, Numer. Linear Algebra Appl., to appear] is based on results contained in this paper. [ABSTRACT FROM AUTHOR]

The graph of linear extensions G(P) of a poset P has as vertices the linear extensions of P, and two vertices are adjacent if they differ only by an adjacent transposition. This graph has so far been investigated mainly with respect to Hamilton paths and intrinsic geodesic convexity. The work on the latter topic especially shows how the graph-theoretic properties of G(P) reflect the order-theoretic structure of P. The aim of this paper is to study the graph G(P) with respect to topics from classical graph theory, e.g., connectivity, cycle space, isometric embeddings, and to find order-theoretic interpretations of these notions. The main theorems in this paper are the equality of the connectivity of G(P) and the jump number of P, the existence of a certain generating system for the cycle space, and a relationship of P and its subposets obtained via an embedding of the graph of linear extensions. [ABSTRACT FROM AUTHOR]

A plane graph G is entirely κ-colorable if V (G) ∪ E(G) ∪ F(G) can be colored with κ colors such that any two adjacent or incident elements receive different colors. In 1993, Borodin proved that every plane graph G with maximum degree Δ ≥ 12 is entirely (Δ + 2)-colorable. In this paper, we improve this result by showing that every plane graph G with Δ ≥ 9 is entirely (Δ + 2)-colorable. [ABSTRACT FROM AUTHOR]

Let G =(V1, V2, E) be a balanced bipartite graph with 2n vertices. The bipartite binding number of G, denoted by B(G), is defined to be n if G = Kn,n and mini∈{1,2} min... |N(S)|/|S| otherwise. We call G bipancyclic if it contains a cycle of every even length m for 4 m 2n. The purpose of this paper is to show that if B(G) > 3/2 and n ≥ 139, then G is bipancyclic; the bound 3/2 is best possible in the sense that there exist infinitely many balanced bipartite graphs G that have B(G)=3/2 but are not Hamiltonian. [ABSTRACT FROM AUTHOR]

In this paper we show that, with 11 exceptions, any matching covered bipartite graph on n vertices, with minimum degree greater than two, has at least 2n - 4 perfect matchings. Using this bound, which is the best possible, and McCuaig's theorem [W. McCuaig, J. Graph Theory, 38 (2001), pp. 124-169] on brace generation, we show that any brace on n vertices has at least (n-2)²/8 perfect matchings. A bi-wheel on n vertices has (n - 2)²/4 perfect matchings. We conjecture that there exists an integer such that every brace on n ≥ N vertices has at least (n-2)²/4 perfect matchings. [ABSTRACT FROM AUTHOR]

An old problem raised independently by Jacobson and Schönheim seeks to determine the maximum for which every graph with edges contains a pair of edge-disjoint isomorphic subgraphs with s edges. In this paper we determine this maximum up to a constant factor. We show that every m-edge graph contains a pair of edge-disjoint isomorphic subgraphs with at least c(m log m)2/3 edges for some absolute constant c, and find graphs where this estimate is off only by a multiplicative constant. Our results improve bounds of Erdös, Pach, and Pyber from 1987. [ABSTRACT FROM AUTHOR]

A graph G is called (r, s)-edge-choosable if for every assignment of sets of size r to the edges of G it is possible to choose for every edge an s-element subset from its set such that the subsets chosen for any pair of adjacent edges are disjoint. The list fractional chromatic index of a graph G is the infimum of all numbers r/s for which G is (r, s)-edge-choosable. Mohar [http://www.fmf.uni-lj.si/~mohar/(June 2003)] posed a problem asking whether every cubic graph is (7,2)-edge-choosable. In 2009, Cranston and West [SIAM J. Discrete Math., 23 (2009), pp. 872-881] showed that every 3-edge-colorable cubic graph is (7,2)-edge-choosable and gave a sufficient condition with the help of which they proved that many non-3-edge-colorable-cubic graphs are (7,2)-edge-choosable. In this paper we prove that every bridgeless cubic graph is (7,2)-edge-choosable. We show that this result cannot be improved in the family of all cubic graphs, in the sense that there exists a cubic graph with list fractional chromatic index 7/2. The original question of Mohar remains open, and we further pose several related problems. [ABSTRACT FROM AUTHOR]

A graph G of order n is said to be ∑' labeled if there exist a bijection f : V (G) → [n] and a constant c such that for all v ∈ V(G), ∑u∈N[v]f(u) = c. The uniqueness of the constant c has been an open question that was posed by Prof. Arumugam at the 2010 IWOGL Conference in Duluth. The question of the uniqueness of such a constant can be extended to arbitrary neighborhoods and to arbitrary sets of labels. For a set D ⊂ N, let ND(v) = {u ∈ V(G): d(u, v) ∈ D}, and let W be a multiset of real numbers. Graph G is said to be (D, W)-vertex magic if there exists a bijection g : V(G) → W such that for all v ∈ V(G), ∑u∈ND(v)g(u) is a constant, called the (D, W)-vertex magic constant. In this paper we prove that, even for these more general conditions, a (D, W)-vertex magic constant will be unique. Moreover, the constant can be determined using a generalization of the fractional domination number of the graph. [ABSTRACT FROM AUTHOR]

In this paper, we construct new families of graphs whose automorphism groups are transitive on 3-paths. These graphs are constructed from certain Lie algebras related to generalized Kac-Moody algebras of rank two. We will show that one particular subfamily gives new lower bounds on the number of edges in extremal graphs with no cycles of length fourteen. [ABSTRACT FROM AUTHOR]

Let Δ (G) be the maximum degree of a graph G. Brooks' theorem states that the only connected graphs with chromatic number X(G) = Δ (G) + 1 are complete graphs and odd cycles. We prove a fractional analogue of Brooks' theorem in this paper. Namely, we classify all connected graphs G such that the fractional chromatic number Xf(G) is at least Δ (G). These graphs are complete graphs, odd cycles, C²8 C5 ⊠ K2, and graphs whose clique number ω(G) equals the maximum degree Δ (G). Among the two sporadic graphs, the graph C²8 is the square graph of cycle C8, while the other graph C5 ⊠ K2 is the strong product of C5 and K2. In fact, we prove a stronger result: If a connected graph G with δ (G) ≥ 4 is not one of the graphs listed above, then we have Xf(G)≤ Δ(G) -- 2/67. [ABSTRACT FROM AUTHOR]

Thomassen [Combinatorica, 3 (1983), pp. 393--396] proved that every digraph with minimum out-degree at least three has two vertex disjoint cycles. There are examples of 3-regular digraphs where all pairs of vertex disjoint cycles have the same length. In this paper we raise the conjectures that all 3-regular bipartite digraphs and all digraphs with minimum degree at least four have two vertex disjoint cycles of different length. We give support for our conjectures by proving that all 4-regular digraphs do indeed have two vertex disjoint cycles of different length. We furthermore discuss consequences of our results and conjectures as well as arc-weighted versions of our conjecture [ABSTRACT FROM AUTHOR]

Let G = (V, E) be a graph. A set S⊆V is independent if no two vertices from are adjacent, while core(G) is the intersection of all maximum independent sets [V. E. Levit and E. Mandrescu, Discrete Appl. Math., 117 (2002), pp. 149-161]. The independence number α(G)is the cardinality of a largest independent set, and µ(G) is the size of a maximum matching of G. The neighborhood of A⊆V is N(A)= {v ∈ V : N(v)∩=0. The number dc(G)=max{|X|-|N(X)| : X ⊆ V} is called the critical di?erence of G, and is critical if |A|-|N(A)| = dc(G) [C. Q. Zhang, SIAM J. Discrete Math., 3 (1990), pp. 431-438]. We define ker(G) as the intersection of all critical sets. In this paper we prove that if dc(G) ≥ 1, then ker(G) ⊆ core(G) and |ker(G)| >dc(G) ≥ α(G)-µ(G). [ABSTRACT FROM AUTHOR]

In this short paper we study the game of cops and robbers, which is played on the vertices of some fixed graph G. Cops and a robber are allowed to move along the edges of G, and the goal of cops is to capture the robber. The cop number c(G) of G is the minimum number of cops required to win the game. Meyniel conjectured a long time ago that O(√n) cops are enough for any connected G on n vertices. Improving several previous results, we prove that the cop number of an n-vertex graph is at most n2-(1+0(1))√log n.A similar result independently and slightly before us was also obtained by Lu and Peng. [ABSTRACT FROM AUTHOR]

In this paper, we consider the half-integral disjoint paths packing. For a graph G and k pairs of vertices (s1, t1), (s2, t2), …, (sk, tk) in G, the objective is to find paths P1, …, Pk in G such that Pi joins si and ti for i = 1, 2, …, k, and in addition, each vertex is on at most two of these paths. We give a polynomial-time algorithm to decide the feasibility of this problem with k = O((log n/log log n)1/7). This improves a result by Kleinberg [Proceedings of the 30th ACM Symposium on Theory of Computing, 1998, pp 530-539] who proved the same conclusion when k = O((log log n)2/15). Our algorithm still works for several problems related to the bounded unsplittable flow. These results can all carry over to problems involving edge capacities. Our main technical contribution is to give a "crossbar" of a polynomial size of the tree width of the graph. [ABSTRACT FROM AUTHOR]

In multiflow maximization problems, there are several combinatorial duality relations, such as the Ford-Fulkerson max-flow min-cut theorem for single commodity flows, Hu's max-biflow min-cut theorem for two-commodity flows, the Lovász-Cherkassky duality theorem for free multiflows, and so on. In this paper, we provide a unified framework for such multiflow combinatorial dualities by using the notion of a folder complex, which is a certain 2-dimensional polyhedral complex introduced by Chepoi. We show that for a nonnegative weight μ on terminal set, the μ-weighted maximum multiflow problem admits a combinatorial duality relation if and only if μ is represented by distances between certain subsets in a folder complex, and we show that the corresponding combinatorial dual problem is a discrete location problem on the graph of the folder complex. This extends a result of Karzanov in the case of metric weights. [ABSTRACT FROM AUTHOR]

We study the maximum edge-disjoint paths problem in undirected planar graphs: given a graph G and node pairs (demands) s1t1, s2t2, … , sktk, the goal is to maximize the number of demands that can be connected (routed) by edge-disjoint paths. The natural multicommodity flow relaxation has an Ω(√n) integrality gap, where n is the number of nodes in G. Motivated by this, we consider solutions with small constant congestion c > 1, that is, solutions in which up to c paths are allowed to use an edge (alternatively, each edge has a capacity of c). In previous work we obtained an O(log n) approximation with congestion 2 via the flow relaxation. This was based on a method of decomposing into well-linked subproblems. In this paper we obtain an O(1) approximation with congestion 4. To obtain this improvement we develop an alternative decomposition that is specific to planar graphs. The decomposition produces instances that we call Okamura-Seymour (OS) instances. These have the property that all terminals lie on a single face. Another ingredient we develop is a constant factor approximation for the all-or-nothing flow problem on OS instances via the flow relaxation. [ABSTRACT FROM AUTHOR]

We study the convergence of additive and multiplicative Schwarz iterations applied to singular M-matrices and Markov chains. We do our investigations in order to solve consistent linear systems or to calculate a probability distribution vector of a Markov chain. It turns out that for a certain set of equations we are able to prove convergence for both methods with a reliable degree of freedom concerning the overlap. These new convergence theorems are based on a graph theoretical approach and represent the main results of this paper. Other applications of the introduced theory are also discussed. [ABSTRACT FROM AUTHOR]

The first-fit chromatic number of a graph is the number of colors needed in the worst case of a greedy coloring. It is also called the Grundy number, which is defined to be the maximum number of classes in an ordered partition of the vertex set of a graph G into independent sets V1, V2, … , Vk so that for each 1 ≤ i < j ≤ k and for each x ϵ Vi there exists a y ϵ Vi such that x and y are adjacent. In this paper, we study the first-fit chromatic number of outerplanar and planar graphs as well as Cartesian products of graphs, and in particular we give asymptotically tight results for outerplanar graphs. [ABSTRACT FROM AUTHOR]

Two graphs G and H on n vertices are ϵ-far from being isomorphic if at least ϵ(n 2) edges must be added or removed from E(G) in order to make G and H isomorphic. In this paper we deal with the question of how many queries are required to distinguish between the case that two graphs are isomorphic and the case that they are ϵ-far from being isomorphic. A query is defined as probing the adjacency matrix of any one of the two graphs, i.e., asking if a pair of vertices forms an edge of the graph or not. We investigate both one-sided and two-sided error testers under two possible settings: The first setting is where both graphs need to be queried, and the second setting is where one of the graphs is fully known to the algorithm in advance. We prove that the query complexity of the best one-sided error testing algorithm is Θ(n3/2) if both graphs need to be queried, and that it is Θ(n) if one of the graphs is known in advance (where the Θ notation hides polylogarithmic factors in the upper bounds). For two-sided error testers, we prove that the query complexity of the best tester is Θ(√n) when one of the graphs is known in advance, and we show that the query complexity lies between Θ(n) and Õ(n5/4) if both G and H need to be queried. All of our algorithms are additionally nonadaptive, while all of our lower bounds apply for adaptive testers as well as nonadaptive ones. [ABSTRACT FROM AUTHOR]

This paper studies an impulsive dynamical system that is in the form of a measuredriven differential inclusion. The employed solution concept depends upon a graph completion of the measure. The first main result shows that a subsequence of discrete-time trajectories graph-converges to a solution, and, under Lipschitz hypotheses, that every solution can be obtained in this manner. The second main result pursues a similar development with classical systems that approximate the given one. [ABSTRACT FROM AUTHOR]

The strong chromatic index of a graph $G$, denoted by $\chi_s(G)$, is the minimum number of colors needed to color its edges so that each color class is an induced matching. In this paper we analyze the asymptotic behavior of this parameter in a random graph $G(n,p)$, for two regions of the edge probability $p=p(n)$. For the dense case, where $p$ is a constant, $0 < p < 1$, we prove that with high probability $\chi_s(G)\le (1+o(1))\frac{3}{4}\frac{n^2p}{\log_bn}$, where $b=1/(1-p)$. This improves upon a result of Czygrinow and Nagle [{\it Discrete Math.}, 281 (2004), pp. 129--136]. For the sparse case, where $np< \frac{1}{100}\sqrt{\log n/\log\log n}$, we show that with high probability $\chi_s(G)=\Delta_1(G)$, where $\Delta_1(G)=\max\{d(u)+d(v)-1:\ (u,v)\in E(G)\}$. This improves a result of Palka [{\it Australas. J. Combin.}, 18 (1998), pp. 219--226]. [ABSTRACT FROM AUTHOR]

The decycling number $\nabla(G)$ of a graph $G$ is the smallest number of vertices which can be removed from $G$ so that the resultant graph contains no cycles. In this paper, we study the decycling number for the family of graphs consisting of the Cartesian product of two cycles. We completely solve the problem of determining the decycling number of $C_m \square C_n$ for all $m$ and $n$. Moreover, we find a vertex set $T$ that yields a maximum induced tree in $C_m\square C_n$. [ABSTRACT FROM AUTHOR]

For a graph $G$, let $\diam(G)$ denote the diameter of $G$. For any two vertices $u$ and $v$ in $G$, let $d(u, v)$ denote the distance between $u$ and $v$. A multilevel distance labeling (or distance labeling) for $G$ is a function $f$ that assigns to each vertex of $G$ a nonnegative integer such that for any vertices $u$ and $v$, $|f(u)-f(v)| \geq \diam(G) - d_G(u, v) +1$. The span of $f$ is the largest number in $f(V)$. The radio number of $G$, denoted by $rn(G)$, is the minimum span of a distance labeling for $G$. In this paper, we completely determine the radio numbers for paths and cycles. [ABSTRACT FROM AUTHOR]

Enumerating the isomorphism classes of several types of graph covering projections is one of the central research topics in enumerative topological graph theory (see [S. F. Du, D. Marusic, and A. O. Waller, J. Combin. Theory Ser. B, 74 (1998), pp. 276--290], [S. F. Du, J. H. Kwak, and M. Y. Xu, J. Combin. Theory Ser. B, 93 (2005), pp. 73--93], [R. Feng, J. H. Kwak, J. Kim, and J. Lee, SIAM J. Discrete Math., 11 (1998), pp. 265--272], [R. Feng. and J. H. Kwak, Discrete Math.}, 277 (2004), pp. 73--85], [C. D. Godsil and A. D. Hensel, J. Combin. Theory Ser. B., 56 (1992), pp. 205--238], [M. Hofmeister, Discrete Math., 143 (1995), pp. 87--97], [M. Hofmeister, SIAM J. Discrete Math., 8 (1995), pp. 51--61], [M. Hofmeister, SIAM J. Discrete Math., 11 (1998), pp. 286--292], [J. H. Kwak, J. Chun, and J. Lee, SIAM J. Discrete Math., 11 (1998), pp. 273--285], [J. H. Kwak and J. Lee, Canad. J. Math., 42 (1990), pp. 747--761], and [J. H. Kwak and J. Lee, Combinatorial and Computational Mathematics: Present and Future, (2001), pp. 97--161]). A covering projection is called circulant if its covering graph is circulant. A covering projection p from a Cayley graph ${\rm Cay} ({\cal A},X)$ onto another ${\rm Cay} ({\cal Q},Y)$ is called typical if the map $p: {\cal A}\rightarrow {\cal Q}$ on the vertex sets is a group homomorphism from ${\cal A}$ onto ${\cal Q}$. In [R. Feng. and J. H. Kwak, Discrete Math., 277 (2004), pp. 73--85], the authors enumerated the isomorphism classes of typical circulant double covering projections onto a circulant graph. As a continuation of this work, we enumerate in this paper the isomorphism classes of those covering projections of any folding number. [ABSTRACT FROM AUTHOR]

Recently, D. Corneil found a simple 3-sweep lexicographic breadth first search (LexBFS) algorithm for the recognition of proper interval graphs. We point out how to modify Corneil's algorithm to make it a certifying algorithm, and then describe a similar certifying 3-sweep LexBFS algorithm for the recognition of proper interval bigraphs. It follows from an earlier paper that the class of proper interval bigraphs is equal to the better known class of bipartite permutation graphs, and so we have a certifying algorithm for that class as well. All our algorithms run in time O(m+n), including the certification phase. The certificates of representability (the intervals) can be authenticated in time O(m + n). The certificates of nonrepresentability (the forbidden subgraphs) can be authenticated in time O(n). [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of testing bipartiteness of general graphs. The problem has previously been studied in two models, one most suitable for dense graphs and one most suitable for bounded-degree graphs. Roughly speaking, dense graphs can be tested for bipartiteness with constant complexity, while the complexity of testing bounded-degree graphs is [This symbol cannot be presented in ASCII format](√n), where n is the number of vertices in the graph (and [This symbol cannot be presented in ASCII format](f(n)) means Θ(f(n) · polylog(f(n)))). Thus there is a large gap between the complexity of testing in the two cases. In this work we bridge the gap described above. In particular, we study the problem of testing bipartiteness in a model that is suitable for all densities. We present an algorithm whose complexity is [This symbol cannot be presented in ASCII format](min(√n, n²/m)), where m is the number of edges in the graph, and we match it with an almost tight lower bound. [ABSTRACT FROM AUTHOR]

Splitting off two edges su, sv in a graph G means deleting su, sv and adding a new edge uv. Let G = (V + s, E) be k-edge-connected in V (k ≥ 2) and let d(s) be even. Lovász proved that the edges incident to s can be split off in pairs in a such a way that the resulting graph on vertex set V is k-edge-connected. In this paper we investigate the existence of such complete splitting sequences when the set of split edges has to meet additional requirements. We prove structural properties of the set of those pairs u, v of neighbors of s for which splitting off su, sv destroys k-edge-connectivity. This leads to a new method for solving problems of this type. By applying this method we obtain a short proof for a recent result of Nagamochi and Eades on planarity-preserving complete splitting sequences and prove the following new results: let G and H be two graphs on the same set V + s of vertices and suppose that their sets of edges incident to s coincide. Let G (H) be k-edge-connected (l-edge-connected, respectively) in V (k, l ≥ 2) and let d(s) be even. Then there exists a pair su, sv which can be split off in both graphs preserving k-edge-connectivity in G (l-edge-connectivity in H, respectively), provided d(s) ≥ 6. If k and l are both even, then such a pair always exists. By using these edge-splitting results and the polymatroid intersection theorem we give a polynomial algorithm for the problem of simultaneously augmenting the edge-connectivity of two graphs by adding a (common) set of new edges of (almost) minimum size. [ABSTRACT FROM AUTHOR]

This paper gives a new and faster algorithm to find a 1-factor in a bipartite Δ-regular graph. The time complexity of this algorithm is O(nΔ + n log n log Δ), where n is the number of nodes. This implies an O(n log n log Δ + m log Δ) algorithm to edge-color a bipartite graph with n nodes, in edges, and maximum degree Δ. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected uniquely vertex k-colorable graph, or a k-UCG for short. M. Truszczyński [Some results on uniquely colorable graphs, in Finite and Infinite Sets, North-Holland, Amsterdam, 1984, pp. 733–748] introduced e*(G) = ¦ V (G) ¦ (k − 1) − k2 as the minimum number of edges for a k-UCG and S. J. Xu [J. Combin. Theory Ser. B, 50 (1990), pp. 319–320] conjectured that any minimal k-UCG contains a Kk as a subgraph. In this paper, first we introduce a technique called forcing. Then by applying this technique in conjunction with a feedback structure we construct a k-UCG with clique number k − t, for each t ≥ 1 and each k, when k is large enough. This also improves some known results for the case t = 1. Second, we analyze the parameter Λ(G) = ¦ E(G) ¦ − e*(G) for our constructions, and we obtain some bounds for the functions λt(k) = min{Λ(G) : G is a k-UCG and cl(G) = k - t}, vt(k) = min{¦ V (G) ¦ : G is a k-UCG and cl(G) = k − t}. Also, we introduce some possible applications of the technique in cryptography and data compression. [ABSTRACT FROM AUTHOR]

Consider the 2n (or O(n²)) min-cut problems on a graph with n nodes and nonnegative edge weights. Gomory and Hu [J. Soc. Indust. Appl. Math., 9 (1961), pp. 551–570] showed (essentially) that there are at most n-1 different min-cuts. They also described a compact structure (the flow equivalent tree) of size O(n) with the following property: for any pair of nodes, the value of a min-cut can be obtained from this structure. Furthermore, they showed how this structure can be found by solving only n - 1 min-cut problems. This paper contains generalizations of these results. For example, consider a k-terminal cut problem on a graph: for a given set of k nodes, delete a minimum weight set of edges (called a k-cut) so that each of the k nodes is in a different component. There are kn (or O(nk)) such problems. Hassin [Math. Oper. Res., 13 (1988), pp. 535–542] showed (essentially) that there are at most k-1n-1 (or O(nk-1)) different min k-cuts. We describe a compact structure of size O(nk-1) with the following property: for any k nodes, the value of a min k-cut can be obtained from this structure. We also show how this structure can be found by solving only k-1n-1 k-terminal cut problems. This work builds upon the results of Hassin [Math. Oper. Res., 13 (1988), pp. 535–542], [Oper. Res. Lett., 9 (1990), pp. 315–318], and [Lecture Notes in Comput. Sci. 450, 1990, pp. 228–299]. [ABSTRACT FROM AUTHOR]

Let G be a 3-connected plane graph. Plummer and Toft [J. Graph Theory, 11 (1987), pp. 507–515] conjectured that χc(G) ≤ Δ&ast;(G) + 2, where χc(G) is the cyclic chromatic number of G and Δ&ast;(G) the maximum face size of G. Horňák and Jendrol' [J. Graph Theory, 30 (1999), pp. 177–189] and Borodin and Woodall [SIAM J. Discrete Math., submitted] independently proved this conjecture when Δ&ast;(G) is large enough. Moreover, Borodin and Woodall proved a stronger statement that χc(G) ≤ Δ&ast;(G) + 1 holds if Δ&ast;(G) ≥ 122. In this paper, we prove that χc(G) ≤ Δ&ast;(G) + 1 holds if Δ&ast;(G) ≥ 60. [ABSTRACT FROM AUTHOR]

A tree, being a connected acyclic graph, can be bicolored in two ways, which differ from each other by exchange of the colors. We shall say that a tree is equicolorable if these bicolorings assign the two colors to equal numbers of vertices. Labelled equicolored trees have been enumerated several times in the literature, and from this result it is easy to enumerate labelled equicolorable trees. The result is that the probability that a randomly chosen n-vertex labelled tree is equicolorable is asymptotically just twice the probability that its vertices would be equicolored if they were assigned colors by independent unbiased coin flips. Our goal in this paper is the enumeration of unlabelled equicolorable trees (that is, trees up to isomorphism), both exactly (in terms of generating functions) and asymptotically. We treat both the rooted and unrooted versions of this problem and conclude that in either case the probability that a randomly chosen n-vertex unlabelled tree is equicolorable is asymptotically 1.40499… times as large as the probability that it would be equicolored if its vertices were assigned colors by independent unbiased coin flips. [ABSTRACT FROM AUTHOR]

In this paper we consider procedures for completely disconnecting a complete graph on n vertices Kn. By this we simply mean that we wish to remove all of the edges of Kn in a sequence of steps by which the graph Kn is successively transformed into the empty graph on n vertices. The rules are that, on each step, we may remove at most one edge from each connected component of the present graph, and in addition we may impose a limit w on the maximum number of edges we can remove at a time. What is the minimum number of steps fw (n) it will take to completely disconnect Kn? The two cases considered are w = ∞ and w = 2. We show that the ratio of fw(n) to the number of edges of Kn is asymptotic to 2/3 in the first case and to 1/2 + λ(1 - λ) = .74194412 … in the second, where λ is the real root of the equation λ = 2(1 - λ)³. [ABSTRACT FROM AUTHOR]

The compactness of a graph measures the space complexity of its shortest path routing tables. Each outgoing edge of a node x is assigned a (pairwise disjoint) set of addresses, such that the unique outgoing edge containing the address of a node y is the first edge of a shortest path from x to y. The complexity measure used in the context of interval routing is the minimum number of intervals of consecutive addresses needed to represent each such set, minimized over all possible choices of addresses and all choices of shortest paths. This paper establishes asymptotically tight bounds of n/4 on the compactness of an n-node graph. More specifically, it is shown that every nnode graph has compactness at most n/4 + o(n), and conversely, there exists an n-node graph whose compactness is n/4 - o(n). Both bounds improve upon known results. (A preliminary version of the lower bound has been partially published in Proceedings of the 22nd International Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Comput. Sci. 1300, pp. 259-268, 1997. [ABSTRACT FROM AUTHOR]

Bender, Richmond, and Wormald showed that in almost all planar 3-connected triangulations (or dually, 3-connected cubic maps) with n edges, the largest 4-connected triangulation (or dually, the largest cyclically 4-edge-connected cubic component) has about n/2 edges [Random Structures Algorithms, 7 (1995), pp. 273-285]. In this paper, we derive some general results about the size of the largest component and apply them to a variety of types of planar maps. [ABSTRACT FROM AUTHOR]

Ear decomposition of a graph has been extensively studied in relation to graph connectivity. In this paper, a connection of ear decomposition to graph embeddings is exhibited. It is shown that constructing a maximum-paired ear decomposition of a graph and constructing a maximum-genus embedding of the graph are polynomial-time equivalent. Applications of this connection are discussed. [ABSTRACT FROM AUTHOR]

In the well-solved edge-connectivity augmentation problem we must find a minimum cardinality set F of edges to add to a given undirected graph to make it k-edge-connected. This paper solves the generalization where every edge of F must go between two different sets of a given partition of the vertex set. A special case of this partition-constrained problem, previously unsolved, is increasing the edge-connectivity of a bipartite graph to k while preserving bipartiteness. Based on this special case we present an application of our results in statics. Our solution to the general partition-constrained problem gives a min-max formula for F which includes as a special case the original min-max formula of Cai and Sun [Networks, 19 (1989), pp. 151-172] for the problem without partition constraints. When k is even the rain-max formula for the partition-constrained problem is a natural generalization of the unconstrained version. However, this generalization fails when k is odd. We show that at most one more edge is needed when k is odd and we characterize the graphs that require such an extra edge. We give a strongly polynomial algorithm that solves our problem in time O(n(m + n log n) log n). Here n and m denote the number of vertices and distinct edges of the given graph, respectively. This bound is identical to the best-known time bound for the problem without partition constraints. Our algorithm is based on the splitting off technique of Lovász, like several known efficient algorithms for the unconstrained problem. However, unlike previous splitting algorithms, when k is odd our algorithm must handle obstacles that prevent all edges from being split off. Our algorithm is of interest even when specialized to the unconstrained problem, because it produces an asymptotically optimum number of distinct splits. [ABSTRACT FROM AUTHOR]

The following constrained matching problem arises in the area of manpower scheduling. Consider an undirected graph G = (V, E) and a digraph D = (V, A). A master/slave-matching (MS-matching) in G with respect to D is a matching in G such that for each arc (u,v) ∈ A for which the node u is matched, the node v is matched too. The problem is to find an MS-matching of maximum cardinality. This paper addresses the special case where G is bipartite with bipartition V = W ∪ U and every (weakly) connected component of D is either an isolated node or two nodes in U which are joined by a single arc. The polyhedral structure of this special case is investigated and a min-max theorem which characterizes the cardinality of a maximum MS-matching in terms of the weight of a special node cover is derived. This min-max theorem includes as a special case the theorem of König. [ABSTRACT FROM AUTHOR]

In this paper we describe new constructions for de Bruijn sequences and Perfect Factors. These constructions are all based upon the idea of constructing one sequence (or set of sequences) from another. As a result of this fact, the sequences obtained from these construction methods possess simple decoding algorithms based on decoding the sequences used to construct them. Such decoding algorithms are of importance in position-location applications. [ABSTRACT FROM AUTHOR]

The article discusses the paper "The Alcuin Number of a Graph and Its Connections to the Vertex Cover Number" by Péter Csorba, Cor A. J. Hurkens and Gerhard J. Woeginger. The paper discusses a mathematical problem raised by scholar, monk and teacher Alcuin of York. It provides examples of reachability problems from computer science, operations research and puzzles.

In mathematics, problems with exact solutions are like gold nuggets. They often allow one to illustrate the fundamental behavior of a class of problems with elegance and simplicity, and at a level the neophyte can understand. One such nugget is described in this issue's Education paper, "Critical Percolation on a Bethe Lattice Revisited" by Braga, Sanchis, and Schieber. Percolation theory lives at the intersection of graph theory and probability. The following is a typical percolation theory question: Given a porous material (such as a sponge) with a set of pores connected by channels that are randomly chosen to be open or closed, what is the probability that a pore in the center is connected to the outside boundary? (That is, what is the probability that fluid can percolate from the outside boundary to the central pore?) Questions of this type arise in numerous applications, including modeling the spread of an epidemic or a forest fire, the flow of groundwater, and the formation of polymer networks. Our paper considers percolation theory on a Bethe lattice (of which a binary tree is perhaps the most familiar example). The authors build the theory from the ground up and describe a beautiful example of what physicists call a phase transition and the associated concepts of critical probability and critical exponents. Moreover, in this model the authors describe how to compute these quantities exactly. The paper also illustrates how percolation theory can become particularly subtle as the domain size becomes infinite. This material could naturally be introduced into the undergraduate curriculum as a case-study in a discrete mathematics or a graph theory course. The paper also may kindle interest in what is an active and applied area of graduate study and research. On a more personal note, this issue commences my term as Editor of the Education section. Fortunately, my predecessor, Bobby Schnabel, is not far away - I wish him every success in his new role as Editor-in-Chief of SIAM Review. [ABSTRACT FROM AUTHOR]