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In this paper we prove several results on Ramsey numbers R(H,F) for a fixed graph H and a large graph F, in particular for F = Kn. These results extend earlier work of Erd\H os, Faudree, Rousseau, and Schelp and of Balister, Schelp, and Simonovits on so-called Ramsey sizelinear graphs. Among other results, we show that if H is a subdivision of K4 with at least six vertices, then R(H,F) =O(v(F)+e(F)) for every graph F. We also conjecture that if H is a connected graph with e(H) v(H) (k+1/2)2, then R(H,Kn) = O(nk). The case k = 2 was proved by Erd\H os, Faudree, Rousseau, and Schelp. We prove the case k = 3. [ABSTRACT FROM AUTHOR]

A remarkable conjecture of Feige [SIAM J. Comput., 35 (2006), pp. 964-984] asserts that for any collection of n independent nonnegative, where X = Σ ni=1=i. In this paper, we investigate thisconjecture for the class of discrete log-concave probability distributions, and we prove a strengthened version. More specifically, we show that the conjectured bound 1/e holds when Xi's are independent discrete log-concave with arbitrary expectation. [ABSTRACT FROM AUTHOR]

The famous Erdős-Sós conjecture states that every graph of average degree more than t -- 1 must contain every tree on t + 1 vertices. In this paper, we study a spectral version of this conjecture. For n > k, let Sn,k be the join of a clique on fc vertices with an independent set of n -- k vertices and denote by Sf k the graph obtained from Sn>k by adding one edge. We show that for fixed k > 2 and sufficiently large n, if a graph on n vertices has adjacency spectral radius at least as large as Sn,k and is not isomorphic to Sn,k, then it contains all trees on 2k + 2 vertices. Similarly, if a sufficiently large graph has spectral radius at least as large as S+n,k then it either contains all trees on 2k + 3 vertices or is isomorphic to S+n,k. This answers a two-part conjecture of Nikiforov affirmatively. [ABSTRACT FROM AUTHOR]

Let n, s, and k be positive integers such that k ≥ 3, s ≥ 3, and n ≥ ks. An smatchingMs in a k-uniform hypergraph is a set of s pairwise disjoint edges. The anti-Ramsey number ar(n, k,Ms) of an s-matching is the smallest integer c such that each edge-coloring of the n-vertex k-uniform complete hypergraph with exactly c colors contains an s-matching with distinct colors. In 2013, Özkahya and Young proposed a conjecture on the exact value of ar(n, k,Ms) for all n ≥ sk and k ≥ 3. A 2019 result by Frankl and Kupavskii verified this conjecture for all n ≥ sk + (s 1)(k 1) and k ≥ 3. We aim to determine the value of ar(n, 3,Ms) for 3s ≤ n < 5s 2 in this paper. Namely, we prove that if 3s < n < 5s-2 and n is large enough, then ar(n, 3,Ms) = ex(n, 3,Ms 1) + 2. Here ex(n, 3,Ms 1) is the Turán number of an (s 1)-matching. Thus this result confirms the conjecture of Özkahya and Young for k = 3, 3s < n < 5s 2 and sufficiently large n. For n = ks and k ≥ 3, we present a new construction for the lower bound of ar(n, k,Ms) which shows the conjecture by Özkahya and Young is not true. In particular, for n = 3s, we prove that ar(n, 3,Ms) = ex(n, 3,Ms-1) + 5 for sufficiently large n. [ABSTRACT FROM AUTHOR]

Assume λ = { k1, k2,..., kq} is a partition of kλ = sum Σqi=1 ki. A λ-list assignment of G is a kλ -list assignment L of G such that the color set... can be partitioned into | λ | =q sets C1,C2, . . . ,Cq such that for each i and each vertex v of...≥ ki. We say G is λ -choosable if G is L-colorable for any λ -list assignment L of G. The concept of λ -choosability is a refinement of choosability that puts k-choosability and k-colorability in the same framework. If | λ | is close to kλ, then λ -choosability is close to kλ -colorability; if | λ | is close to 1, then λ -choosability is close to kλ -choosability. This paper studies Hadwiger's conjecture in the context of λ -choosability. Hadwiger's conjecture is equivalent to saying that every Kt-minor-free graph is { 1 (t 1)} -choosable for any positive integer t, where { 1 (t 1)} is the multiset consisting of t 1 copies of 1. We prove that for t geqslant 5, for any partition λ of t 1 other than { 1 (t 1)}, there is a Kt-minor-free graph G that is not λ -choosable. We then construct several types of Kt-minor-free graphs that are not λ -choosable, where kλ (t 1) gets larger as kλ | λ | gets larger. In particular, for any q and any ε > 0, there exists t0 such that for any t geqslant t0, for any partition λ of lfloor (2 ε)trfloor with | λ | = q, there is a Kt-minor-free graph that is not λ -choosable. The q = 1 case of this result was recently proved by Steiner, and our proof uses a similar argument. We also generalise this result to (a, b)-list coloring. [ABSTRACT FROM AUTHOR]

Let C2 be the cyclic group of order 2 and D2n be the dihedral group of order 2n, where n is even. In this paper, we provide the exact values of some zero-sum constants over D2n × C2, namely the small Davenport constant, Gao constant, n--constant and Erd\H os--Ginzburg--Ziv constant. As a consequence, we prove Gao's conjecture and the Zhuang--Gao conjecture for this group. These are the first concrete results on zero-sum problems for a family of non-abelian groups of rank greater than two. [ABSTRACT FROM AUTHOR]

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]

Fuglede's conjecture states that for a subset Ω of a locally compact abelian group G with positive and finite Haar measure, there exists a subset of the dual group of G which is an orthogonal basis of L² (Ω) if and only if it tiles the group by translations. In this paper, we prove a divisibility property for a set in Zp × Zpn. Then, using the divisibility property and equidistribution property, we prove that Fuglede's conjecture holds in the group Zp × Zpn. [ABSTRACT FROM AUTHOR]

In the classical model of network creation games introduced by Fabrikant et al. [On a network creation game, in Proceedings of the Twenty-Second Annual Symposium on Principles of Distributed Computing (PODC'03), 2003, pp. 347-351], players correspond to the nodes of a network buying links of price and to the other players with the goal of being well-connected to the resulting network. Still as an open problem, the constant PoA conjecture states that the Price of Anarchy (PoA) is constant for any. When tackling this problem distinct behaviors must be taken into the account depending on whether has either large or low value. It is known that for every ne is a tree and for with the diameter of networks that are in equilibrium when restricting to deviations that consist only in buying links (buying equilibria) is at most a constant. These results imply that the PoA is constant for the disjoint union of the two ranges, and thus the constant PoA conjecture seems to be true for most of all the possible values. In this paper we study the PoA for the remaining range of and we show the following: (i) For the PoA is constant by proving that the size of any biconnected component of an equilibrium graph is constant. (ii) For we have that, where is the maximum diameter of an equilibrium graph for the same range of. Therefore if the constant PoA conjecture was false, it would suffice to construct equilibria of nonconstant diameter. Towards this direction we find nontrivial buying equilibria of nonconstant diameter when and, exploring new intimate relationships between distance-uniform graphs and buying equilibria. [ABSTRACT FROM AUTHOR]

Let G be a 2-connected n-vertex graph and Ns(G) be the total number of s-cliques in G. Let k=4 and s=2 be integers. In this paper, we show that if G has an edge e which is not on any cycle of length at least k, then Ns(G)=r(k-1s)+(t+2s), where n-2=r(k-3)+t and 0=t=k-4. This result settles a conjecture of Ma and Yuan and provides a clique version of a theorem of Fan, Wang and Lv. As a direct corollary, if Ns(G)>r(k-1s)+(t+2s), every edge of G is covered by a cycle of length at least k. [ABSTRACT FROM AUTHOR]

In 1990, Hendry conjectured that all Hamiltonian chordal graphs are cycle extendable. After a series of papers confirming the conjecture for a number of graph classes, the conjecture is yet refuted by Lafond and Seamone in 2015. Given that their counterexamples are not strongly chordal graphs and they are all only 2-connected, Lafond and Seamone asked the following two questions: (1) Are Hamiltonian strongly chordal graphs cycle extendable? (2) Is there an integer k such that all k-connected Hamiltonian chordal graphs are cycle extendable? Later, a conjecture stronger than Hendry's is proposed. In this paper, we resolve all these questions in the negative. On the positive side, we add to the list of cycle-extendable graphs two more graph classes, namely, Hamiltonian 4-fan-free chordal graphs, where every induced K5 e has true twins, and Hamiltonian \{ 4-fan,A\} -free chordal graphs. [ABSTRACT FROM AUTHOR]

It is conjectured that every bridgeless graph G has a family \scrF of even subgraphs (cycles) such that each edge of G is contained in precisely two members of \scrF (CDC), and a stronger conjecture says that every bridgeless graph has a 5-even subgraph double cover (5-CDC). Both the CDC and the 5-CDC conjectures are confirmed for cubic graphs with oddness at most 4. In this paper, we first introduce a new approach related to nowhere-zero 4-flows to attack these two conjectures, and then apply it together with Catlin reduction to present a sufficient condition for a superposition to have a 5-CDC. As an application of those results, the 5-CDC conjecture is verified for several families of snarks. [ABSTRACT FROM AUTHOR]

A matching of a graph is a set of edges without common end vertex. A graph is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. Recently, Biedl and Wittnebel [J. Graph Theory, 99 (2022), pp. 217--230] proved that every 1-planar graph with minimum degree 3 and n \geq 7 vertices has a matching of size at least n+12 7, which is tight for some graphs. They also provided tight lower bounds for the sizes of matchings in 1-planar graphs with minimum degree 4 or 5. In this paper, we show that any 1-planar graph with minimum degree 6 and n \geq 36 vertices has a matching of size at least 3n+4 7, and this lower bound is tight. Our result confirms a conjecture posed by Biedl and Wittnebel [J. Graph Theory, 99 (2022), pp. 217--230]. [ABSTRACT FROM AUTHOR]

Let r \geq k \geq 2 and K(k) r,n denote the complete n-balanced r-partite k-uniform hypergraph, whose vertex set consists of r parts, each has n vertices, and whose edge set contains all the k-element subsets with no two vertices from one part. A decomposition of K(k) r,n is a partition of E(K(k) r,n). A perfect matching (resp., Hamilton tight cycle) decomposition of K(k) r,n is a decomposition of K(k) r,n into perfect matchings (resp., Hamilton tight cycles). In this paper, we prove that if k | n (resp., 2 - k and k | n), then K(k) k+1,n (resp., K(k) k+2,n) has a perfect matching decomposition. We also prove that for any integer k \geq 2, K(k) k+1,n has a Hamilton tight cycle decomposition. In all cases, we use constructive methods involving number theory. In fact, we confirm two conjectures proposed by Zhang, Lu, and Liu [Appl. Math. Comput., 386 (2020), 125492]. [ABSTRACT FROM AUTHOR]

For any positive integer $k$, define $f(k)$ (respectively, $f_3(k)$) to be the minimal integer $\ge k$ such that every 3-connected planar graph $G$ (respectively, 3-connected cubic planar graph $G$) of circumference $\ge k$ has a cycle whose length is in the interval $[k, f(k)]$ (respectively, $[k, f_3(k)]$). Merker showed that $f_3(k) \le 2k + 9$ for any $k \ge 2$, and $f_3(k) \ge 2k + 2$ for any even $k \ge 4$. He conjectured that $f_3(k) \le 2k + 2$ for any $k \ge 2$. This conjecture was disproved by Zamfirescu, who gave an infinite family of counterexamples for every even $k \ge 6$ whose graphs have no cycle length in $[k, 2k + 2]$, i.e., $f_3(k) \ge 2k + 3$ for any even $k \ge 6$. However, the exact value of $f_3(k)$ was only known for $k \le 4$, and it is a natural problem to determine $f_3(k)$ for $k \ge 5$. In this paper, we improve Merker's upper bound, and give the exact value of $f_3(k)$ for every $k \ge 5$. We show that $f_3(5) = 10$, $f_3(7) = 15$, $f_3(9) = 20$, and $f_3(k) = 2k + 3$ for any $k = 6, 8$ or $\ge 10$. For general 3-connected planar graphs, Merker conjectured that there exists some positive integer $c$ such that $f(k) \le 2k + c$ for any positive integer $k$. We give a complete positive answer to this conjecture. We prove that $f(k) = 5$ for any $k \le 3$, $f(4) = 10$, and $f(k) = 2k + 3$ for any $k \ge 5$. [ABSTRACT FROM AUTHOR]

A long-standing conjecture of Zsolt Tuza asserts that the triangle covering number \tau (G) is at most twice the triangle packing number \nu (G), where the triangle packing number \nu (G) is the maximum size of a set of edge-disjoint triangles in G and the triangle covering number \tau (G) is the minimal size of a set of edges intersecting all triangles. In this paper, we prove that Tuza's conjecture holds in the Erd\H os--R\'enyi random graph G(n,m) for all ranges of m, closing the "gap" in what was previously known. (Recently, this result was also independently proved by Jeff Kahn and Jinyoung Park.) We employ a random greedy process called the online triangle packing process to produce a triangle packing in G(n,m) and analyze this process by using the differential equations method. [ABSTRACT FROM AUTHOR]

Let C2 be the cyclic group of order 2 and D2n be the dihedral group of order 2n, where n is even. In this paper, we provide the exact values of some zero-sum constants over D2n × C2, namely the small Davenport constant, Gao constant, n--constant and Erd\H os--Ginzburg--Ziv constant. As a consequence, we prove Gao's conjecture and the Zhuang--Gao conjecture for this group. These are the first concrete results on zero-sum problems for a family of non-abelian groups of rank greater than two. [ABSTRACT FROM AUTHOR]

Let PG(k) be the number of proper k-colorings of a finite simple graph G. Tomescu's conjecture, which was recently solved by Fox, He, and Manners, states that PG(k) leq k!(k 1)n k for all connected graphs G on n vertices with chromatic number kgeq 4. In this paper, we study the same problem with the additional constraint that G is ell -connected. For 2-connected graphs G, we prove a tight bound PG(k) leq (k 1)!((k 1)n k+1 + (1)n k) and show that equality is only achieved if G is a k-clique with an ear attached. For ell geq 3, we prove an asymptotically tight upper bound PG(k) leq k!(k 1)n ell k+1 + O((k 2)n) and provide a matching lower bound construction. For the ranges kgeqll or ellgeq (k 2)(k 1) + 1 we further find the unique graph maximizing PG(k). We also consider generalizing ell -connected graphs to connected graphs with minimum degree delta. [ABSTRACT FROM AUTHOR]

In this paper we describe a parallel algorithm for generating all nonisomorphic rank 3 simple matroids with a given multiplicity vector. We apply our implementation in the high performance computing version of GAP to generate all rank 3 simple matroids with at most 14 atoms and an integrally splitting characteristic polynomial. We have stored the resulting matroids alongside with various useful invariants in a publicly available, ArangoDB-powered database. As a byproduct we show that the smallest divisionally free rank 3 arrangement which is not inductively free has 14 hyperplanes and exists in all characteristics distinct from 2 and 5. Another database query proves that Terao's freeness conjecture is true for rank 3 arrangements with 14 hyperplanes in any characteristic. [ABSTRACT FROM AUTHOR]

Whitney proved in 1931 that 4-connected planar triangulations are Hamiltonian. Hakimi, Schmeichel, and Thomassen conjectured in 1979 that if G is a 4-connected planar triangulation with n vertices, then G contains at least 2(n 2)(n 4) Hamiltonian cycles, with equality if and only if G is a double wheel. On the other hand, a recent result of Alahmadi, Aldred, and Thomassen states that there are exponentially many Hamiltonian cycles in 5-connected planar triangulations. In this paper, we consider 4-connected planar n-vertex triangulations G that do not have too many separating 4-cycles or have minimum degree 5. We show that if G has O(n/log2n) separating 4-cycles, then G has \Omega (n2) Hamiltonian cycles, and if \delta (G) \geq 5, then G has 2\Omega (n1/4) Hamiltonian cycles. Both results improve previous work. Moreover, the proofs involve a "double wheel" structure, providing further evidence to the above conjecture. [ABSTRACT FROM AUTHOR]

In this paper we show examples for applications of the Bombieri--Lang conjecture in additive combinatorics, giving bounds on the cardinality of sumsets of squares and higher powers of integers. [ABSTRACT FROM AUTHOR]

Fuglede's conjecture states that for a subset Ω of a locally compact abelian group G with positive and finite Haar measure, there exists a subset of the dual group of G which is an orthogonal basis of L² (Ω) if and only if it tiles the group by translations. In this paper, we prove a divisibility property for a set in Zp × Zpn. Then, using the divisibility property and equidistribution property, we prove that Fuglede's conjecture holds in the group Zp × Zpn. [ABSTRACT FROM AUTHOR]

How many cliques can a graph on n vertices have with a forbidden substructure Extremal problems of this sort have been studied for a long time. This paper studies the maximum possible number of cliques in a graph on n vertices with a forbidden clique subdivision or immersion. We prove for t sufficiently large that every graph on n geq t vertices with no Kt-immersion has at most n2t+log22 t cliques, which is sharp apart from the 2O(log2 t) factor. We also prove that the maximum number of cliques in an n-vertex graph with no Kt-subdivision is at most 21.817tn for sufficiently large t. This improves on the best known exponential constant by Lee and Oum. We conjecture that the optimal bound is 32t/3+o(t)n, as we proved for minors in place of subdivision in earlier work. [ABSTRACT FROM AUTHOR]

In a recent paper, Garber, Gavrilyuk, and Magazinov [Discrete Comput. Geom., 53 (2015), pp. 245{260] proposed a sucient combinatorial condition for a parallelohedron to be anely Voronoi. We show that this condition holds for all 5-dimensional Voronoi parallelohedra. Consequently, the Voronoi conjecture in R5 holds if and only if every 5-dimensional parallelohedron is combinatorially Voronoi. Here, by saying that a parallelohedron P is combinatorially Voronoi, we mean that P is combinatorially equivalent to a Dirichlet{Voronoi polytope for some lattice Λ, and this combinatorial equivalence is naturally translated into equivalence of the tiling by copies of P with the Voronoi tiling of Λ. We also propose a new condition which, if satisfied by a parallelohedron P, is sufficient to infer that P is affinely Voronoi. The condition is based on the new notion of the Venkov complex associated with a parallelohedron and cohomologies of this complex. [ABSTRACT FROM AUTHOR]

For a collection of N unit vectors X = { xi} N i=1, define the p-frame energy of X as the quantity sum inot =j | langle xi, xj rangle | p. In this paper, we connect the problem of minimizing this value to another optimization problem, thus giving new lower bounds for such energies. In particular, for p < 2, we prove that this energy is at least 2(N d)p p 2 (2 p) p 2 2 which is sharp for d leq N leq 2d and p = 1. We also prove that for 1 leq m < d, a repeated orthonormal basis construction of N = d + m vectors minimizes the energy over an interval, p in [1, pm], and demonstrates an analogous result for all N in the case d = 2. Finally, we give conjectures on these and other energies. [ABSTRACT FROM AUTHOR]

Given a graph H, the Tur'an number ex(n,H) is the largest number of edges in an Hfree graph on n vertices. We make progress on a recent conjecture of Conlon, Janzer, and Lee [More on the Extremal Number of Subdivisions, arXiv:1903.10631v1, 2019] on the Tur'an numbers of bipartite graphs, which in turn yields further progress on a conjecture of Erdos and Simonovits [Combinatorica, 1 (1981), pp. 25-42]. Let s, t, k = 2 be integers. Let Kk s,t denote the graph obtained from the complete bipartite graph Ks,t by replacing each edge uv in it with a path of length k between u and v such that the st replacing paths are internally disjoint. It follows from a general theorem of Bukh and Conlon [J. Eur. Math. Soc. (JEMS), 20 (2018), pp. 1747-1757] that ex(n,Kk s,t) = O(n1+ 1k - 1 sk). Conlon, Janzer, and Lee recently conjectured that for any integers s, t, k = 2, ex(n,Kk s,t) = O(n1+ 1k - 1 sk). Among many other things, they settled the k = 2 case of their conjecture. As the main result of this paper, we prove their conjecture for k = 3, 4. Our main results also yield infinitely many new so-called Tur'an exponents: rationals r ? (1, 2) for which there exists a bipartite graph H with ex(n,H) = T(nr), adding to the lists recently obtained by Jiang, Ma, and Yepremyan [On Tur'an Exponents of Bipartite Graphs, arXiv:1806.02838, 2018], by Kang, Kim, and Liu [On the Rational Tur'an Exponent Conjecture, arXiv:1811.06916, 2018], and by Conlon, Janzer, and Lee. Our method builds on an extension of the Conlon-Janzer-Lee method. We also note that the extended method also gives a weaker version of the Conlon-Janzer-Lee conjecture for all k = 2. [ABSTRACT FROM AUTHOR]

For a graph F, the k-subdivision of F, denoted Fk, is the graph obtained by replacing the edges of F with internally vertex-disjoint paths of length k. In this paper, we prove that ex(n,Kk s,t) = O(n1+ s 1 sk), which is tight for t sufficiently large. This settles a conjecture of Conlon-- Janzer--Lee, and improves on a substantial body of work by Conlon--Janzer--Lee and Jiang--Qiu. [ABSTRACT FROM AUTHOR]

Let k be a positive integer. Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least 2k 1 contains k vertex-disjoint cycles. This conjecture is famous as one of a hundred unsolved problems selected in [A. Bondy and M. R. Murty, Graph Theory, Springer-Verlag, London, 2008]. Lichiardopol, P\'or, and Sereni proved in [SIAM J. Discrete Math., 23 (2009), pp. 979--992] that the above conjecture holds for k = 3. Let g be the girth, i.e., the length of the shortest cycle, of a given digraph. Bang-Jensen, Bessy, and Thomass\'e conjectured in [J. Graph Theory, 75 (2014), pp. 284--302] that every digraph with girth g and minimum outdegree at least g g 1 k contains k vertex-disjoint cycles. Thomass\'e conjectured around 2006 that every oriented graph (a digraph without 2-cycles) with girth g and minimum outdegree at least h contains a path of length h(g 1), where h is a positive integer. In this paper, we first present a new shorter proof of the Bermond--Thomassen conjecture for the case of k = 3, and then we disprove the conjecture proposed by Bang-Jensen, Bessy, and Thomass\'e. Finally, we disprove the even girth case of the conjecture proposed by Thomass\'e. [ABSTRACT FROM AUTHOR]

Hajós conjectured in 1961 that every k-chromatic graph contains a Kk-subdivision. In this paper, we consider the subdivision of K5- and prove that every 4-connected graph contains a K5- -subdivision. This may make progress for the case k = 5 of the Hajós' conjecture. [ABSTRACT FROM AUTHOR]

Let ℤ+ be the set of positive integers. Let Ck denote all subsets of ℤ+ such that none of them contains k + 1 pairwise coprime integers and Ck(n) = Ck ᑎ{ 1, 2, . . ., n\} . Let f(n, k) = maxA∈Ck(n)| A|, where | A| denotes the number of elements of the set A. Let Ek(n) be the set of positive integers not exceeding n which are divisible by at least one of the primes p1, . . ., pk, where pi denotes the ith prime number. In 1962, Erdős conjectured that f(n, k) = | E(n, k)| for every n ≥ pk. Recently Chen and Zhou proved some results about this conjecture. In this paper we solve an open problem of Chen and Zhou and prove several related results about the conjecture. [ABSTRACT FROM AUTHOR]

For a subset A of the set of positive integers, a graph G is called A-coverable if G has a cycle (a subgraph in which all vertices have even degree) which intersects all edge-cuts T in G with |T| ∈ A, and A is said to be coverable if all graphs are A-coverable. As a possible approach to the dominating cycle conjecture, Kaiser and Škrekovski conjectured in [SIAM J. Discrete Math., 22 (2008), pp. 861-874] that ℕ + 3 is coverable, where ℕ + 3 = {4, 5, 6,...}. In this paper, we disprove Kaiser and Škrekovski's conjecture by showing that there exist infinitely many graphs which are not {4, 5}-coverable. [ABSTRACT FROM AUTHOR]

The routing number rt(G) of a connected graph G is the minimum integer r so that every permutation of vertices can be routed in r steps by swapping the ends of disjoint edges. In this paper, we study the routing numbers of cycles, complete bipartite graphs, and hypercubes. We prove that rt(Cn) = n - 1 (for n ≥ 3) and for Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. We also prove n + 1 ≤ rt(Qn) ≤ 2n - 2 for n ≥ 3. The lower bound rt(Qn) > ≥ + 1 was previously conjectured by Alon, Chung, and Graham [SIAM J. Discrete Math., 7 (1994), pp. 513-530]. A variation, called fractional routing number, is also considered in this paper. [ABSTRACT FROM AUTHOR]

A famous conjecture of Pósa from 1962 asserts that every graph on n vertices and with minimum degree at least 2n=3 contains the square of a Hamilton cycle. The conjecture was proven for large graphs in 1996 by Komlós, Sárközy, and Szemerédi [Random Structures Algorithms, 9 (1996) pp. 193-211]. In this paper we prove a degree sequence version of Pósa's conjecture: Given any η > 0, every graph G of sufficiently large order n contains the square of a Hamilton cycle if its degree sequence d1 ≤ ... ≤ dn satisfies di ≤ (1/3 + η)n + i for all i ≤ n/3. The degree sequence condition here is asymptotically best possible. Our approach uses a hybrid of the regularity-blowup method and the connecting-absorbing method. [ABSTRACT FROM AUTHOR]

could benefit from a unilateral deviation. We consider the simplest case of the parallel links network, where links are related. The Social Cost of a Nash equilibrium is the expected maximum latency. We seek the worst-case Nash equilibrium [E. Koutsoupias and C. H. Papadimitriou, Comput. Sci. Rev., 3 (2009), pp. 65-69], which maximizes Social Cost. We continue the study of the fully mixed Nash equilibrium conjecture, abbreviated as the FMNE Conjecture, stating that the worst-case Nash equilibrium is the fully mixed Nash equilibrium, where each user assigns strictly positive probability to every link. Through an extensive combinatorial analysis, we confirm the FMNE Conjecture for the two basic cases where there are either (i) two users on related links, or (ii) many users on two identical links. [ABSTRACT FROM AUTHOR]

Tuza [Finite and Infinite Sets, Proc. Colloq. Math. Soc. János Bolyai 37, North Holland, 1981, p. 888] conjectured that τ(G) ≤ 2v(G) for all graphs G, where τ(G) is the minimum size of an edge set whose removal makes G triangle-free and v(G) is the maximum size of a collection of pairwise edge-disjoint triangles. Here, we generalize Tuza's conjecture to simple binary matroids that do not contain the Fano plane as a restriction and prove that the geometric version of the conjecture holds for cographic matroids. [ABSTRACT FROM AUTHOR]

We show that for every r ≥ 3, the maximal running time of the Kr+1r-bootstrap percolation in the complete r-uniform hypergraph on n vertices Knr is ϴ(nr). This answers a recent question of Noel and Ranganathan in the affirmative and disproves a conjecture of theirs. Moreover, we show that the prefactor is of the form r-reO(r) as r → ∞, [ABSTRACT FROM AUTHOR]

The perfect matching index of a cubic graph G, denoted by \pi (G), is the smallest number of perfect matchings that cover all the edges of G. According to the Berge--Fulkerson conjecture, \pi (G) \leq 5 for every bridgeless cubic graph G. The class of graphs with \pi \geq 5 is of particular interest as many conjectures and open problems, including the famous cycle double cover conjecture, can be reduced to it. Although nontrivial examples of such graphs are very difficult to find, a few infinite families are known, all with circular flow number \Phi c(G) = 5. It has been therefore suggested [Abreu et al., Electron. J. Combin., 23 (2016), P3.54] that \pi (G) \geq 5 might imply \Phi c(G) \geq 5. In this article we dispel these hopes and present a family of cyclically 4-edgeconnected cubic graphs of girth at least 5 with \pi \geq 5 and \Phi c \leq 4 + 2 3. [ABSTRACT FROM AUTHOR]

M. A. Henning and A. Yeo conjectured in [SIAM J. Discrete Math., 26 (2012), pp. 687{694] that a bipartite digraph of minimum out-degree at least 3 contains two vertex disjoint directed cycles of different lengths. In this paper, we disprove this conjecture. Further, we classify strong tournaments and strong bipartite tournaments of minimum out-degree 3 without two vertex disjoint directed cycles of different lengths. [ABSTRACT FROM AUTHOR]

A sequence π = (d1, . . . , dn) is graphic if there is a simple graph G with vertex set {v1, . . . , vn} such that the degree of vi is di. We say that graphic sequences π1 = (d(1)1 , . . . , d(1)n ) and π2 = (d(2)1 , . . . , d(2)n ) pack if there exist edge-disjoint n-vertex graphs G1 and G2 such that for j ∈ {1, 2}, dGj(vi) = d(j)i i for all i ∈ {1, . . . ,n}. Here, we prove several extremal degree sequence packing theorems that parallel central results and open problems from the graph packing literature. Specifically, the main result of this paper implies degree sequence packing analogues to the Bollobás– Eldridge–Catlin graph packing conjecture and the classical graph packing theorem of Sauer and Spencer. In discrete tomography, a branch of discrete imaging science, the goal is to reconstruct discrete objects using data acquired from low-dimensional projections. Specifically, in the k-color discrete tomography problem the goal is to color the entries of an m × n matrix using k colors so that each row and column receives a prescribed number of entries of each color. This problem is equivalent to packing the degree sequences of k bipartite graphs with parts of sizes m and n. Here we also prove several Sauer–Spencer-type theorems with applications to the two-color discrete tomography problem. [ABSTRACT FROM AUTHOR]

We prove that every connected P5-free graph has cop number at most two, solving a conjecture of Sivaraman. In order to do so, we first prove that every connected P5-free graph G with independence number at least three contains a three-vertex induced path with vertices a-b-c in order, such that every neighbor of c is also adjacent to one of a, b. [ABSTRACT FROM AUTHOR]

For an oriented graph D and a set X \subseteq V (D), the inversion of X in D is the digraph obtained by reversing the orientations of the edges of D with both endpoints in X. The inversion number of D, inv(D), is the minimum number of inversions which can be applied in turn to D to produce an acyclic digraph. Answering a recent question of Bang-Jensen, da Silva, and Havet we show that, for each k ∈ N and tournament T, the problem of deciding whether inv(T)≤k is solvable in time Ok(| V (T)| ²), which is tight for all k. In particular, the problem is fixed-parameter tractable when parameterized by k. On the other hand, we build on their work to prove their conjecture that for k \geq 1 the problem of deciding whether a general oriented graph D has inv(D) ≤k is NP-complete. We also construct oriented graphs with inversion number equal to twice their cycle transversal number, confirming another conjecture of Bang-Jensen, da Silva, and Havet, and we provide a counterexample to their conjecture concerning the inversion number of so-called dijoin digraphs while proving that it holds in certain cases. Finally, we asymptotically solve the natural extremal question in this setting, improving on previous bounds of Belkhechine, Bouaziz, Boudabbous, and Pouzet to show that the maximum inversion number of an n-vertex tournament is (1 +o(1))n. [ABSTRACT FROM AUTHOR]

A graph drawn in the plane is called k-quasi-planar if it does not contain k pair-wise crossing edges. It has been conjectured for a long time that for every fixed k, the maximum number of edges of a k-quasi-planar graph with n vertices is O(n). The best known upper bound is n(logn)O(logk). In the present paper, we improve this bound to (n log n)2α(n)ck in the special case where the graph is drawn in such a way that every pair of edges meet at most once. Here α(n) denotes the (extremely slowly growing) inverse of the Ackermann function. We also make further progress on the conjecture for k-quasi-planar graphs in which every edge is drawn as an x-monotone curve. Extending some ideas of Valtr, we prove that the maximum number of edges of such graphs is at most 2ck6 n log n. [ABSTRACT FROM AUTHOR]

Reed's w, Δ, χ conjecture proposes that every graph satisfies χ ≤ [1/2(Δ + 1 + w)] it is known to hold for all claw-free graphs. In this paper we consider a local strengthening of this conjecture. We prove the local strengthening for line graphs, then note that previous results immediately tell us that the local strengthening holds for all quasi-line graphs. Our proofs lead to polytime algorithms for constructing colorings that achieve our bounds: O(n²) for line graphs and O(n³m²) for quasi-line graphs. For line graphs, this is faster than the best known algorithm for constructing a coloring that achieves the bound of Reed's original conjecture. [ABSTRACT FROM AUTHOR]

A Roman dominating function of a graph G is a function f: V (G) → {0, 1, 2} such that whenever f(v) = 0, there exists a vertex u adjacent to v such that f(u)=2 . T h e weight of f is w(f)= f(v). Σ∈V(G) f(v). The Roman domination number γR (G) of G is the minimum weight of a Roman dominating function of G. Chambers, Kinnersley, Prince, and West [ SIAM J. Discrete Math., 23 (2009), pp. 1575-1586] conjectured that γR(G) ≤ [2n/3] for any 2-connected graph G of n vertices. This paper gives counterexamples to the conjecture and proves that γR(G) ≤ max{[2n/3], 23n/34} for any 2-connected graph G of n vertices. We also characterize 2-connected graphs G for which γR(G)=23n/34 when 23n/34 > 2n/3. [ABSTRACT FROM AUTHOR]

Wegner conjectured that for each planar graph G with maximum degree Δ at least 4, Χ(G²) ≤ Δ + 5 if 4 & le; Δ ≤ 7, and Χ(G²) ≤ [3Δ/2] =1 if Δ ≥ 8. Let G be a planar graph without 4- and 5-cycles. In this paper, we discuss the L(p, q)-labeling of G and show that λp, q (G) ≤ (2q-1)Δ+6p+6q-6a and λp, q (G) ≤ max{(2q-1)Δ+6p+2q-4, 9(2q-1)+8p-4, 6(2q-1)+10p-5}, where p and q are positive integers with p ≥ q. As a corollary, Χ(G²) ≤ Δ +7 if Δ 7, Χ(G²) = 14 ≤ 14 if Δ = 8, and Χ(G²) ≤ Δ +5 if Δ ≥ 9. [ABSTRACT FROM AUTHOR]

Thomassen (J. Combin. Theory Ser. B, 62 (1994), pp. 180-181) proved that every planar graph is 5-choosable. This result was generalized by Škrekovski (Discrete Math., 190 (1998), pp. 223-226) and He, Miao, and Shen (Discrete Math., 308 (2008), pp. 4024-4026), who proved that every K5-minor-free graph is 5-choosable. Both proofs rely on the characterization of K5-minor-free graphs due to Wagner (Math. Ann., 114 (1937), pp. 570-590). This paper proves the same result without using Wagner's structure theorem or even planar embeddings. Given that there is no structure theorem for graphs with no K6-minor, we argue that this proof suggests a possible approach for attacking the Hadwiger Conjecture. [ABSTRACT FROM AUTHOR]

The firefighter problem is the following discrete-time game on a graph. Initially, a fire starts at a vertex of the graph. In each round, a firefighter protects one vertex not yet on fire, and then the fire spreads to all unprotected neighbors of the vertices on fire. The objective of the firefighter is to save as many vertices as possible. The surviving rate of a graph is the average percentage of vertices that can be saved when a fire starts randomly at one vertex of the graph, which measures the defense ability of a graph as a whole. In this paper, we study the surviving rates of graphs with bounded treewidth. We prove that the surviving rate of every n-vertex outerplanar graph is at least 1 - Θ ( log n/n), which is asymptotically tight. We also prove that if k firefighters are available in each round, then the surviving rate of an n-vertex graph with treewidth at most k is at least 1 - O(k ² log n/n)- Furthermore, we show that the greedy strategy of Hartnell and Li [Congr. Numer., 145 (2000), pp. 187-192] for trees saves at least 1 - Θ (log n/n) percent of vertices on average for an n-vertex tree. Our results settle a conjecture and two problems of Cai and Wang [SIAM J. Discrete Math., 23 (2009), pp. 1814-1826] in affirmative. [ABSTRACT FROM AUTHOR]

Sorting permutations by transpositions is an important and difficult problem in genome rearrangements. The transposition diameter TD (n) is the maximum transposition distance among all pairs of permutations in Sn. It was previously conjectured [H. Ericksson et al., Discrete Math., 241 (2001), pp. 289-300] that TD (n) ≤ Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. This conjecture was disproved by Elias and Hartman [IEEE/ACM Trans. Comput. Biol. Bioinform.. 3 (2006), 369-379] by showing TD (n) ≥ Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. In this paper we improved the lower bound to TD (n) ≥ 17/33n + 1/33 via computation. [ABSTRACT FROM AUTHOR]

In the present paper we show that if A is a set of n real numbers and the product set A.A has at most n1+ϵ elements, then the h-fold sumset hA has at least nlog(h/2)/2 log 2+1/2-fh(ϵ) elements, where fh(c) → 0 as c → 0. We also prove results on the h-fold sumset h(A.A) = A.A + ⋯ + A.A. [ABSTRACT FROM AUTHOR]

In this paper, we show that every planar graph without 7-cycles is 4-choosable. [ABSTRACT FROM AUTHOR]