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Let G be a graph. A 2-rainbow dominating function (or 2-RDF) of G is a function f from V (G) to the set of all subsets of the set f1; 2g such that for a vertex v 2 V (G) with f(v) =;, the condition S u2NG(v) f(u) = f1; 2g is fulfilled, where NG(v) is the open neighborhood of v. The weight of 2-RDF f of G is the value !(f):= P v2V (G) jf(v)j. The 2-rainbow domination number of G, denoted by r2(G), is the minimum weight of a 2-RDF of G. A 2-RDF f is called an outer independent 2-rainbow dominating function (or OI2-RDF) of G if the set of all v 2 V (G) with f(v) =; is an independent set. The outer independent 2-rainbow domination number oir2(G) is the minimum weight of an OI2-RDF of G. In this paper, we obtain the outer independent 2-rainbow domination number of PmPn and PmCn. Also we determine the value of oir2(Cm2Cn) when m or n is even. [ABSTRACT FROM AUTHOR]

The Szeged index is a modification of the Wiener index to cyclic molecules. The Szeged index of a connected graph G is defined as SZ(G)= Σe∈E(G) n1(e|G)n2(e|G), where E(G) is the edge set of G, and for any e = uν ∈ E(G), n1(e|G) is the number of vertices of G lying closer to vertex u than to vertex ν, and n2(e|G) is the number of vertices of G lying closer to vertex ν than to vertex u. In this paper, we determine the n-vertex unicyclic graphs whose vertices on the unique cycle have degree at least three with the first, the second and the third smallest as well as largest Szeged indices for n ≥ 6, n ≥ 7 and n ≥ 8, respectively. [ABSTRACT FROM AUTHOR]

Abstract We consider the class of Berge graphs that do not contain an induced cycle of length four. We present a purely graph-theoretical algorithm that produces an optimal coloring in polynomial time for every graph in that class. [ABSTRACT FROM AUTHOR]

Abstract The balanced hypercube was proposed as a novel interconnection network for large-scale parallel systems. It is known that the balanced hypercube is edge-bipancyclic. Given a set of edges P with | P | ≤ n − 1 and two vertices x and y in different bipartite sets, we show that there exists a Hamiltonian path from x to y passing through P in B H n if and only if P is a linear forest and, neither x nor y is an internal vertex of P , and x and y are not end-vertices of a component of P simultaneously. Consequently, the balanced hypercube contains a Hamiltonian cycle passing through a set P of at most n prescribed edges if and only if P is a linear forest. Moreover, our result improves some known results. [ABSTRACT FROM AUTHOR]

Abstract For graphs G and H , let G → H signify that any red/blue edge coloring of G contains a monochromatic H as a subgraph. Let G (K k (n) , p) be random graph spaces with edge probability p , where K k (n) is the complete k -partite graph with n vertices in each part. It is shown that if n p → ∞ , then Pr [ G (K k (n) , p) → P (k − 1 − o (1)) n ] → 1 for k = 2 , 3. [ABSTRACT FROM AUTHOR]

Abstract This paper presents an efficient method for constructing 2 2 n − 2 cyclic different de Bruijn sequences of order 2 n from the feedback function of a de Bruijn sequence of order n. This is done by extending an n -stage nonlinear feedback shift register (NFSR) to a 2 n -stage NFSR, and then recursively joining all cycles of the 2 n -stage NFSR to form a full cycle. [ABSTRACT FROM AUTHOR]

Abstract In 2012, Gutman and Wagner proposed the concept of the matching energy of a graph and pointed out that its chemical applications can go back to the 1970s. The matching energy of a graph G is defined as the sum of the absolute values of the zeros of the matching polynomial of G. In this paper, we first present some new techniques of comparing the matching energies of two graphs. As the applications of these techniques, we can determine the unicyclic graphs of order n with the first to the eighth smallest matching energies for all n ≥ 42. [ABSTRACT FROM AUTHOR]

Abstract According to Richardson's theorem, every digraph G without directed odd cycles that is either (a) locally finite or (b) rayless has a kernel (an independent subset K with an incoming edge from every vertex in G − K). We generalize this theorem showing that a digraph without directed odd cycles has a kernel when (a) for each vertex, there is a finite set separating it from all rays, or (b) each ray contains at most finitely many vertices dominating it (having an infinite fan to the ray) and the digraph has finitely many ends. The restriction to finitely many ends in (b) can be weakened, admitting infinitely many ends with a specific structure, but the possibility of dropping it remains a conjecture. [ABSTRACT FROM AUTHOR]

Abstract Balancedly splittable Hadamard matrices are introduced and studied. A connection is made to the Hadamard diagonalizable strongly regular graphs, maximal equiangular lines set, and unbiased Hadamard matrices. Several construction methods are presented. As an application, commutative association schemes of 4, 5, and 6 classes are constructed. [ABSTRACT FROM AUTHOR]

Abstract The fixing number of a graph Γ is the minimum number of labeled vertices that, when fixed, remove all nontrivial automorphisms from the automorphism group of Γ. The fixing set of a finite group G is the set of all fixing numbers of graphs whose automorphism groups are isomorphic to G. Previously, authors have studied the fixing sets of both abelian groups and symmetric groups. In this article, we determine the fixing set of the dihedral group. [ABSTRACT FROM AUTHOR]

Abstract Every pair of points lying on a polygonal path P in the plane has a detour associated with it, which is the ratio between their distance along the path and their Euclidean distance. Given a set S of points along the path, this information can be encoded in a weighted complete graph on S. Among all spanning trees on this graph, a bottleneck spanning tree is one whose maximum edge weight is minimum. We refer to such a tree as a bottleneck detour tree of S. In other words, a bottleneck detour tree of S is a spanning tree in which the maximum detour (with respect to the original path) between pairs of adjacent points is minimum. We show how to find a bottleneck detour tree in expected O (n log 3 ⁡ n + m) time, where P consists of m edges and | S | = n. [ABSTRACT FROM AUTHOR]

Abstract A celebrated unresolved conjecture of Erdős and Hajnal states that for every undirected graph H there exists ϵ (H) > 0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or stable set of size at least n ϵ (H) . The conjecture has a directed equivalent version stating that for every tournament H there exists ϵ (H) > 0 such that every H -free n -vertex tournament T contains a transitive subtournament of order at least n ϵ (H) . We say that a tournament is prime if it does not have nontrivial homogeneous sets. So far the conjecture was proved only for some specific families of prime tournaments (Berger et al., 2014; Choromanski, 2015 [3]) and tournaments constructed according to the so-called substitution procedure (Alon et al., 2001). In particular, recently the conjecture was proved for all five-vertex tournaments (Berger et al. 2014), but the question about the correctness of the conjecture for all six-vertex tournaments remained open. In this paper we prove that all but at most one six-vertex tournament satisfy the Erdős–Hajnal conjecture. That reduces the six-vertex case to a single tournament. [ABSTRACT FROM AUTHOR]

Abstract Given a minor-closed class A of graphs, let β A denote the supremum over all graphs in A of the ratio of edges to vertices. We investigate the set B of all such values β A , taking further the project begun by Eppstein. Amongst other results, we determine the small values in B (those up to 2); we show that B is 'asymptotically dense'; and we answer some questions posed by Eppstein. [ABSTRACT FROM AUTHOR]

Let G = (V,E) be a graph and S ⊆ V. We say that S is a dominating set of G, if each vertex in V \ S has a neighbor in S. Moreover, we say that S is a connected (respectively, 2-edge connected or 2-connected) dominating set of G if G[S] is connected (respectively, 2-edge connected or 2-connected). The domination (respectively, connected domination, or 2-edge connected domination, or 2-connected domination) number of G is the cardinality of a minimum dominating (respectively, connected dominating, or 2-edge connected dominating, or 2-connected dominating) set of G, and is denoted γ(G) (respectively γ1(G), or γ′ 2(G), or γ2(G)). A well-known result of Duchet and Meyniel states that γ1(G) ≤ 3γ(G) - 2 for any connected graph G. We show that if γ(G) ≥ 2, then γ′ 2(G) ≤ 5γ(G) - 4 when G is a 2-edge connected graph and γ2(G) ≤ 11γ(G) - 13 when G is a 2-connected triangle-free graph. [ABSTRACT FROM AUTHOR]

In this paper, we consider T-colorings of directed graphs. In particular, we consider as a T-set the set Tr = {0, 1, 2, . . ., r-1, r+1, . . .}. Exact values and bounds of the Tr-span of directed graphs whose underlying graph is a wheel graph are presented. [ABSTRACT FROM AUTHOR]

A graph G with p vertices and q edges is said to have Skolem difference Lucas mean labelling if it is possible to label the vertices x v with distinct elements f(x) from the set {1,2,...,Lp+q} in such a way that the edge e = uv is labelled with ...f(v)| is odd, then the resulting edge labels are distinct and are from Skolem difference Lucas mean labelling is called a Skolem difference Lucas mean graph. In this paper, we prove for some graphs such as triangular snake graph are Skolem difference Lucas mean graph. [ABSTRACT FROM AUTHOR]

In this paper we prove that some known graphs such as the Herschel graph and some graphs constructed in this paper are divisor cordial graphs. [ABSTRACT FROM AUTHOR]

We discuss a notion of shuffle for trees which extends the usual notion of a shuffle for two natural numbers. We give several equivalent descriptions, and prove some algebraic and combinatorial properties. In addition, we characterize shuffles in terms of open sets in a topological space associated to a pair of trees. Our notion of shuffle is motivated by the theory of operads and occurs in the theory of dendroidal sets, but our presentation is independent and entirely self-contained. [ABSTRACT FROM AUTHOR]

Let C 1 and C 2 be two sets of cycles. We determine all generalised Ramsey numbers R ( C 1 , C 2 ) such that C 1 or C 2 contains a cycle of length at most 6. This generalises previous results of Erdős, Faudree, Rosta, Rousseau, and Schelp. Furthermore, we give a conjecture for the general case. We also provide a complete classification of most ( C 1 , C 2 ) -critical graphs such that C 1 or C 2 contains a cycle of length at most 5. For length 4, this is an easy extension of a recent result of Wu, Sun, and Radziszowski, in which | C 1 | = | C 2 | = 1 . For lengths 3 and 5, our results are new also in this special case. [ABSTRACT FROM AUTHOR]

We study the activation process in undirected graphs known as bootstrap percolation: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it had at least r active neighbors, for a threshold r that is identical for all vertices. A contagious set is a vertex set whose activation results with the entire graph being active. Let m ( G , r ) be the size of a smallest contagious set in a graph G on n vertices. We examine density conditions that ensure m ( G , r ) = r for all r , and first show a necessary and sufficient condition on the minimum degree. Moreover, we study the speed with which the activation spreads and provide tight upper bounds on the number of rounds it takes until all nodes are activated in such graphs. We also investigate what average degree asserts the existence of small contagious sets. For n ≥ k ≥ r , we denote by M ( n , k , r ) the maximum number of edges in an n -vertex graph G satisfying m ( G , r ) > k . We determine the precise value of M ( n , k , 2 ) and M ( n , k , k ) , assuming that n is sufficiently large compared to k . [ABSTRACT FROM AUTHOR]

We study a version of online edge coloring, where the goal is to color as many edges as possible using only a given number, k, of available colors. All of our results are with regard to competitive analysis. Previous attempts to identify optimal algorithms for this problem have failed, even for bipartite graphs. Thus, in this paper, we analyze even more restricted graph classes, paths and trees. For paths, we consider k=2, and for trees, we consider any k≥2. We prove that a natural greedy algorithm called FIRST-FIT is optimal among deterministic algorithms, on paths as well as trees. For paths, we give a randomized algorithm, which is optimal and better than the best possible deterministic algorithm. For trees, we prove that to obtain a better competitive ratio than FIRST-FIT, the algorithm would have to be both randomized and unfair (i.e., reject edges that could have been colored), and even such algorithms cannot be much better than FIRST-FIT. [ABSTRACT FROM AUTHOR]

Let v > k > i be non-negative integers. The generalized Johnson graph, J ( v , k , i ) , is the graph whose vertices are the k -subsets of a v -set, where vertices A and B are adjacent whenever | A ∩ B | = i . In this article, we derive general formulas for the girth and diameter of J ( v , k , i ) . Additionally, we provide a formula for the distance between any two vertices A and B in terms of the cardinality of their intersection. [ABSTRACT FROM AUTHOR]

We examine the following version of a classic combinatorial search problem introduced by Rényi: Given a finite set X of n elements we want to identify an unknown subset Y of X , which is known to have exactly d elements, by means of testing, for as few as possible subsets A of X , whether A intersects Y or not. We are primarily concerned with the non-adaptive model, where the family of test sets is specified in advance, in the case where each test set is of size at most some given natural number k . Our main results are nearly tight bounds on the minimum number of tests necessary when d and k are fixed and n is large enough. [ABSTRACT FROM AUTHOR]

The Kneser graph K ( n , k ) has as vertices all k -element subsets of [ n ] = { 1 , 2 , … , n } and an edge between any two vertices that are disjoint. If n = 2 k + 1 , then K ( n , k ) is called an odd graph. Let n > 4 and 1 < k < n 2 . In the present paper, we show that if the Kneser graph K ( n , k ) is of even order where n is an odd integer or both of the integers n , k are even, then K ( n , k ) is a vertex-transitive non Cayley graph. Although, these are special cases of Godsil [7] , unlike his proof that uses some very deep group-theoretical facts, ours uses no heavy group-theoretic facts. We obtain our results by using some rather elementary facts of number theory and group theory. We show that ‘almost all’ odd graphs are of even order, and consequently are vertex-transitive non Cayley graphs. Finally, we show that if k > 4 is an even integer such that k is not of the form k = 2 t for some t > 2 , then the line graph of the odd graph O k + 1 is a vertex-transitive non Cayley graph. [ABSTRACT FROM AUTHOR]

Let K = { k 1 , k 2 , … , k r } and L = { l 1 , l 2 , … , l s } be disjoint subsets of { 0 , 1 , … , p − 1 } , where p is a prime and A = { A 1 , A 2 , … , A m } is a family of subsets of [ n ] = { 1 , 2 , … , n } such that | A i | ( mod p ) ∈ K for all A i ∈ A and | A i ∩ A j | ( mod p ) ∈ L for i ≠ j . In 1991, Alon, Babai and Suzuki conjectured that if n ≥ s + max 1 ≤ i ≤ r k i , then | A | ≤ n s + n s − 1 + ⋯ + n s − r + 1 . In 2000, Qian and Ray-Chaudhuri proved the conjecture under the condition n ≥ 2 s − r . In 2015, Hwang and Kim verified this conjecture. In this paper, we will prove that if n ≥ 2 s − 2 r + 1 or n ≥ s + max 1 ≤ i ≤ r k i , then | A | ≤ n − 1 s + n − 1 s − 1 + ⋯ + n − 1 s − 2 r + 1 . This result strengthens both the upper bound of Alon–Babai–Suzuki’s conjecture and Qian and Ray-Chaudhuri’s result, when n ≥ 2 s − 2 . [ABSTRACT FROM AUTHOR]

We study ( a : a ) Maker–Breaker games played on the edge set of the complete graph on n vertices. In the following four games — perfect matching game, Hamilton cycle game, star factor game and path factor game, our goal is to determine the least number of moves which Maker needs in order to win these games. Moreover, for all games except for the star factor game, we show how first player can win in the strong version of these games. [ABSTRACT FROM AUTHOR]

The vertices of Kneser graph K ( n , k ) are the subsets of { 1 , 2 , … , n } of cardinality k , two vertices are adjacent if and only if they are disjoint. The square G 2 of a graph G is defined on the vertex set of G with two vertices adjacent if their distance in G is at most 2. Z. Füredi, in 2002, proposed the problem of determining the chromatic number of the square of the Kneser graph. The first non-trivial problem arises when n = 2 k + 1 . It is believed that χ ( K 2 ( 2 k + 1 , k ) ) = 2 k + c where c is a constant, and yet the problem remains open. The best known upper bounds are by Kim and Park: 8 k ∕ 3 + 20 ∕ 3 for 1 k ≥ 3 (Kim and Park, 2014) and 32 k ∕ 15 + 32 for k ≥ 7 (Kim and Park, 2016). In this paper, we develop a new approach to this coloring problem by employing graph homomorphisms, cartesian products of graphs, and linear congruences integrated with combinatorial arguments. These lead to χ ( K 2 ( 2 k + 1 , k ) ) ≤ 5 k ∕ 2 + c , where c is a constant in { 5 ∕ 2 , 9 ∕ 2 , 5 , 6 } , depending on k ≥ 2 . [ABSTRACT FROM AUTHOR]

The Wiener index W ( G ) of a connected graph G is the sum of distances between all pairs of vertices of G . A connected graph G is said to be a cactus if each of its blocks is either a cycle or an edge. Let G n , t be the set of all n -vertex cacti containing exactly t cycles. Liu and Lu (2007) determined the unique graph in G n , t with the minimum Wiener index. We now establish a sharp upper bound on the Wiener index of graphs in G n , t and identify the corresponding extremal graphs. [ABSTRACT FROM AUTHOR]

We introduce a variant of the vertex-distinguishing edge coloring problem, where each edge is assigned a subset of colors. The label of a vertex is the union of the sets of colors on edges incident to it. In this paper we investigate the problem of finding a coloring with the minimum number of colors where every vertex receives a distinct label. Finding such a coloring generalizes several other well-known problems of vertex-distinguishing colorings in graphs. We show that for any graph (without connected component reduced to an edge or a single vertex), the minimum number of colors for which such a coloring exists can only take 3 possible values depending on the order of the graph. Moreover, we provide the exact value for paths, cycles and complete binary trees. [ABSTRACT FROM AUTHOR]

There exists a significant body of work on determining the acquisition number at(G) of various graphs when the vertices of those graphs are each initially assigned a unit weight. We determine properties of the acquisition number of the path, star, complete, complete bipartite, cycle, and wheel graphs for variations on this initial weighting scheme, with the majority of our work focusing on the expected acquisition number of randomly weighted graphs. In particular, we bound the expected acquisition number E(at(Pn)) of the n-path when n distinguishable "units" of integral weight, or chips, are randomly distributed across its vertices between 0.242n and 0.375n. With computer support, we improve it by showing that E(at(Pn)) lies between 0.29523n and 0.29576n. We then use subadditivity to show that the limiting ratio limE(at(Pn))/n exists, and simulations reveal more exactly what the limiting value equals. The Hoeffding-Azuma inequality is used to prove that the acquisition number is tightly concentrated around its expected value. Additionally, in a different context, we offer a non-optimal acquisition protocol algorithm for the randomly weighted path and exactly compute the expected size of the resultant residual set. [ABSTRACT FROM AUTHOR]

Many problems on graphs can be expressed in the following language: given a graph G = ( V , E ) and a terminal set T ⊆ V , find a minimum size set S ⊆ V which intersects all “structures” (such as cycles or paths) passing through the vertices in T . We refer to this class of problems as terminal set problems. In this paper, we introduce a general method to obtain faster exact exponential time algorithms for several terminal set problems. In the process, we break the O ⁎ ( 2 n ) barrier for the classic Node Multiway Cut , Directed Unrestricted Node Multiway Cut and Directed Subset Feedback Vertex Set problems. [ABSTRACT FROM AUTHOR]

Let G = ( V , E ) be a graph on n vertices and f : V → [ 1 , n ] a one to one map of V onto the integers 1 through n . Let d i l a t i o n ( f ) = max { | f ( v ) − f ( w ) | : v w ∈ E } . Define the bandwidth B ( G ) of G to be the minimum possible value of d i l a t i o n ( f ) over all such one to one maps f . Next define the Kneser Graph K ( n , r ) to be the graph with vertex set [ n ] r , the collection of r -subsets of an n element set, and edge set E = { A B : A , B ∈ [ n ] r , A ∩ B = ∅ } . For fixed r ≥ 4 and n growing we show that B ( K ( n , r ) ) = n − 1 r + 1 2 n − 4 r − 1 − 1 2 n − 1 r − 2 + O ( n r − 4 ) . [ABSTRACT FROM AUTHOR]

A bisection of a graph is a partition of its vertex set into two sets which differ in size by at most 1. In this paper, we consider bisections of directed graphs such that both directions of the directed cuts contain many arcs. Let D be a directed graph with minimum outdegree δ ≥ 2 and m arcs. We show that if δ is even, then D has a bisection V ( D ) = V 1 ∪ V 2 such that min { e ( V 1 , V 2 ) , e ( V 2 , V 1 ) } ≥ ( δ 4 ( δ + 1 ) + o ( 1 ) ) m , where e ( V i , V j ) denotes the number of arcs of D from V i to V j for i ≠ j and i , j = 1 , 2 . If the underlying graph of D does not contain cycles of length 4, then we show that D admits a bisection V ( D ) = V 1 ∪ V 2 such that min { e ( V 1 , V 2 ) , e ( V 2 , V 1 ) } ≥ ( 1 / 4 + o ( 1 ) ) m . [ABSTRACT FROM AUTHOR]

We prove new upper bounds on the multicolour Ramsey numbers of paths and even cycles. It is well known that ( k − 1 ) n + o ( n ) ⩽ R k ( P n ) ⩽ R k ( C n ) ⩽ k n + o ( n ) . The upper bound was recently improved by Sárközy who showed that R k ( C n ) ⩽ k − k 16 k 3 + 1 n + o ( n ) . Here we show R k ( C n ) ⩽ ( k − 1 4 ) n + o ( n ) , obtaining the first improvement to the coefficient of the linear term by an absolute constant. [ABSTRACT FROM AUTHOR]

A family A of sets is said to be intersecting if A ∩ B ≠ ∅ for any A , B ∈ A . Let m , n , k and r be positive integers with m ≥ 2 k ≥ 2 r > n ≥ k . A family F of sets is called an ( m , n , k , r ) - intersecting family if F is an intersecting subfamily of [ m ] k containing { A ∈ [ m ] k : | A ∩ [ n ] | ≥ r } . Maximum ( m , k , k , k ) - and ( m , k + 1 , k , k ) -intersecting families were determined by the well-known Erdős–Ko–Rado theorem and Hilton–Milner theorem, respectively. Recently, Li et al. determined maximum ( m , n , k , k ) -intersecting family when n = 2 k − 1 , 2 k − 2 , 2 k − 3 or m is sufficiently large. In this paper, we determine all the maximum ( m , n , k , r ) -intersecting families. [ABSTRACT FROM AUTHOR]

For integers k , n , c with k , n ≥ 1 , the n -color Rado number R k ( n , c ) is defined to be the least integer N if any, or infinity otherwise, such that for every n -coloring of the set { 1 , 2 , … , N } , there exists a monochromatic solution in that set to the linear equation x 1 + x 2 + ⋯ + x k + c = x k + 1 . A recent conjecture of ours states that R k ( n , c ) should be finite if and only if every divisor d ≤ n of k − 1 also divides c . In this paper, we complete the verification of this conjecture for all k ≤ 7 . As a key tool, we first prove a general result concerning the degree of regularity over subsets of Z of some linear Diophantine equations. [ABSTRACT FROM AUTHOR]

Let G be a graph, and let S be a subset of the vertex set of G . We denote the set of the end vertices of a path P by e n d ( P ) . A path P is an S -path if | V ( P ) | ≥ 2 and V ( P ) ∩ S = e n d ( P ) . An S -path-system is a graph H such that H contains all vertices of S and every component of H is an S -path. In this paper, we give a sharp degree sum condition for a bipartite graph to have a spanning S -path-system. [ABSTRACT FROM AUTHOR]

Let G = ( V , E ) be a graph with vertex set V and edge set E . A vertex v ∈ V v e -dominates every edge incident to it as well as every edge adjacent to these incident edges. The vertex–edge degree of a vertex v is the number of edges v e -dominated by v . Similarly, an edge e = u v e v -dominates the two vertices u and v incident to it, as well as every vertex adjacent to u or v . The edge–vertex degree of an edge e is the number of vertices e v -dominated by edge e . In this paper we introduce these types of degrees and study their properties. [ABSTRACT FROM AUTHOR]

Motivated by a question of Grinblat, we study the minimal number v ( n ) that satisfies the following. If A 1 , … , A n are equivalence relations on a set X such that for every i ∈ [ n ] there are at least v ( n ) elements whose equivalence classes with respect to A i are nontrivial, then A 1 , … , A n contain a rainbow matching, i.e. there exist 2 n distinct elements x 1 , y 1 , … , x n , y n ∈ X with x i ∼ A i y i for each i ∈ [ n ] . Grinblat asked whether v ( n ) = 3 n − 2 for every n ≥ 4 . The best-known upper bound was v ( n ) ≤ 16 n / 5 + O ( 1 ) due to Nivasch and Omri. Transferring the problem into the setting of edge-coloured multigraphs, we affirm Grinblat's question asymptotically, i.e. we show that v ( n ) = 3 n + o ( n ) . [ABSTRACT FROM AUTHOR]

By a finite type-graph we mean a graph whose set of vertices is the set of all k -subsets of [ n ] = { 1 , 2 , … , n } for some integers n ≥ k ≥ 1 , and in which two such sets are adjacent if and only if they realise a certain order type specified in advance. Examples of such graphs have been investigated in a great variety of contexts in the literature with particular attention being paid to their chromatic number. In recent joint work with Tomasz Łuczak, two of the authors embarked on a systematic study of the chromatic numbers of such type-graphs, formulated a general conjecture determining this number up to a multiplicative factor, and proved various results of this kind. In this article we fully prove this conjecture. [ABSTRACT FROM AUTHOR]

A graph G is well-covered if every maximal independent set has the same cardinality q . Let i k ( G ) denote the number of independent sets of cardinality k in G . Brown, Dilcher, and Nowakowski conjectured that the independence sequence ( i 0 ( G ) , i 1 ( G ) , … , i q ( G ) ) was unimodal for any well-covered graph G with independence number q . Michael and Traves disproved this conjecture. Instead they posited the so-called “Roller Coaster” Conjecture: that the terms i ⌈ q 2 ⌉ ( G ) , i ⌈ q 2 ⌉ + 1 ( G ) , … , i q ( G ) could be in any specified order for some well-covered graph G with independence number q . Michael and Traves proved the conjecture for q < 8 and Matchett extended this to q < 12 . In this paper, we prove the Roller Coaster Conjecture using a construction of graphs with a property related to that of having a maximal-clique partition. In particular, we show, for all pairs of integers 0 ≤ k < q and positive integers m , that there is a well-covered graph G with independence number q for which every independent set of size k + 1 is contained in a unique maximal independent set, but each independent set of size k is contained in at least m distinct maximal independent sets. [ABSTRACT FROM AUTHOR]

Given a tree and a set P of non-trivial simple paths on it, Vpt ( P ) is the VPT graph (i.e. the vertex intersection graph) of the paths P of the tree T , and Ept ( P ) is the EPT graph (i.e. the edge intersection graph) of P . These graphs have been extensively studied in the literature. Given two (edge) intersecting paths in a graph, their split vertices are the vertices having degree at least 3 in their union. A pair of (edge) intersecting paths is termed non-splitting if they do not have split vertices (namely if their union is a path). In this work, motivated by an application in all-optical networks, we define the graph Enpt ( P ) of edge-intersecting non-splitting paths of a tree, termed the ENPT graph, as the (edge) graph having a vertex for each path in P , and an edge between every pair of paths that are both edge-intersecting and non-splitting. A graph G is an ENPT graph if there is a tree T and a set of paths P of T such that G = Enpt ( P ) , and we say that 〈 T , P 〉 is a representation of G . We first show that cycles, trees and complete graphs are ENPT graphs. Our work follows the lines of Golumbic and Jamison’s research (Golumbic and Jamison, 1985) in which they defined the EPT graph class, and characterized the representations of chordless cycles (holes). It turns out that ENPT holes have a more complex structure than EPT holes. In our analysis, we assume that the EPT graph corresponding to a representation of an ENPT hole is given. We also introduce three assumptions (P1), (P2), (P3) defined on EPT , ENPT pairs of graphs. In this Part I, using the results of Golumbic and Jamison as building blocks, we characterize (a) EPT , ENPT pairs that satisfy (P1)–(P3), and (b) the unique minimal representation of such pairs. [ABSTRACT FROM AUTHOR]

A k -hypertournament H on n vertices with 2 ≤ k ≤ n is a pair H = ( V , A H ) , where V is a set of n vertices and A H is a set of k -tuples of vertices, called arcs, such that for any k -subset S of V , A H contains exactly one of the k ! k -tuples whose entries belong to S . Obviously, a 2 -hypertournament is a tournament. Moon (1994) proved that for every strong tournament, there is a Hamiltonian cycle which contains at least three pancyclic arcs. In this paper, we will show that for an arbitrary strong k -hypertournament H with n vertices, where 2 ≤ k ≤ n − 2 , there is a Hamiltonian cycle C containing at least three pancyclic arcs, each of which belongs to an m -cycle C m for each m ∈ { 3 , 4 , … , n } such that V ( C 3 ) ⊂ V ( C 4 ) ⊂ ⋯ ⊂ V ( C n ) . [ABSTRACT FROM AUTHOR]

A w -container C w ( u , v ) of a graph G between two distinct vertices u and v is a set of w disjoint paths between u and v . A w -container C w ( u , v ) is a w ∗ -container if every vertex of G is on some path in C w ( u , v ) . A graph G is said to be w ∗ -connected if there exists a w ∗ -container between any two distinct vertices u and v . The connectivity of the arrangement graph A n , k is k ( n − k ) . In this paper, we prove that A n , k is k ( n − k ) ∗ -connected for n − k ≥ 2 . [ABSTRACT FROM AUTHOR]

An edge irregular total k-labeling f : V ∪ E → {1, 2, 3, . . . , k} of a graph G = (V, E) is a labeling of vertices and edges of G in such a way that for any two different edges uv and uv their weights f (u) + f (uv) + f (v) and f (u) + f (uv) + f (v) are distinct. The total edge irregularity strength tes(G) is defined as the minimum k for which the graph G has an edge irregular total k-labeling. In this paper, we determine the total edge irregularity strength of disjoint union of p isomorphic double wheel graphs and disjoint union of p consecutive non-isomorphic double wheel graphs. [ABSTRACT FROM AUTHOR]

Let D be a digraph. A path partition of D is called k -optimal if the sum of the k -norms of its paths is minimal. The k-norm of a path P is min ( | V ( P ) | , k ) . Berge’s path partition conjecture claims that for every k -optimal path partition P there are k disjoint stable sets orthogonal to P . For general digraphs the conjecture has been proven for k = 1 , 2 , λ − 1 , λ , where λ is the length of a longest path in the digraph. In this paper we prove the conjecture for λ − 2 and λ − 3 . [ABSTRACT FROM AUTHOR]

For a given graph G , a nonempty subset S contained in V ( G ) is an alliance iff for each vertex v ∈ S there are at least as many vertices from the closed neighbourhood of v in S as in V ( G ) − S . An alliance is global if it is also a dominating set of G . The alliance partition number of G was defined in Hedetniemi et al. (2004) to be the maximum number of sets in a partition of V ( G ) such that each set is an alliance. Similarly, in Eroh and Gera (2008) the global alliance partition number is defined for global alliances, where the authors studied the problem for (binary) trees. In the paper we introduce a new concept of strategic balance in graphs: for a given graph G , determine whether there is a partition of vertex set V ( G ) into three subsets N , S and I such that both N and S are global alliances. We give a survey of its general properties, e.g., showing that a graph G has a strategic balance iff its global alliance partition number equals at least 2. We construct a linear time algorithm for solving the problem in trees (thus giving an answer to the open question stated in Eroh and Gera (2008)) and studied this problem for many classes of graphs: paths, cycles, wheels, stars, complete graphs and complete k -partite graphs. Moreover, we prove that this problem is NP -complete for graphs with a degree bounded by 4 and state an open question regarding subcubic graphs. [ABSTRACT FROM AUTHOR]

The 3 -path vertex cover problem is an extension of the well-known vertex cover problem. It asks for a vertex set S ⊆ V ( G ) of minimum cardinality such that G − S only contains independent vertices and edges. In this paper we will present a polynomial algorithm which computes two disjoint sets T 1 , T 2 of vertices of G such that (i) for any 3 -path vertex cover S ′ in G [ T 2 ] , S ′ ∪ T 1 is a 3 -path vertex cover in G , (ii) there exists a minimum 3 -path vertex cover in G which contains T 1 and (iii) | T 2 | ≤ 6 ⋅ ψ 3 ( G [ T 2 ] ) , where ψ 3 ( G ) is the cardinality of a minimum 3 -path vertex cover and T 2 is the kernel of G . [ABSTRACT FROM AUTHOR]

The generalized power of a simple graph G , denoted by G k , s , is obtained from G by blowing up each vertex into an s -set and each edge into a k -set, where 1 ≤ s ≤ k 2 . When s < k 2 , G k , s is always odd-bipartite. It is known that G k , k 2 is non-odd-bipartite if and only if G is non-bipartite, and G k , k 2 has the same adjacency (respectively, signless Laplacian) spectral radius as G . In this paper, we prove that, regardless of multiplicities, the H -spectrum of A ( G k , k 2 ) (respectively, Q ( G k , k 2 ) ) consists of all eigenvalues of the adjacency matrices (respectively, the signless Laplacian matrices) of the connected induced subgraphs (respectively, modified induced subgraphs) of G . As a corollary, G k , k 2 has the same least adjacency (respectively, least signless Laplacian) H -eigenvalue as G . We also discuss the limit points of the least adjacency H -eigenvalues of hypergraphs, and construct a sequence of non-odd-bipartite hypergraphs whose least adjacency H -eigenvalues converge to − 2 + 5 . [ABSTRACT FROM AUTHOR]

Inspired by a question of Yannakakis on the Vertex Packing polytope of perfect graphs, we study the Clique–Stable Set separation in a non-hereditary subclass of perfect graphs. A cut ( B , W ) of G (a bipartition of V ( G ) ) separates a clique K and a stable set S if K ⊆ B and S ⊆ W . A Clique–Stable Set separator is a family of cuts such that for every clique K , and for every stable set S disjoint from K , there exists a cut in the family that separates K and S . Given a class of graphs, the question is to know whether every graph of the class admits a Clique–Stable Set separator containing only polynomially many cuts. It was recently proved to be false for the class of all graphs (Göös, 2015), but it remains open for perfect graphs, which was Yannakakis’ original question. Here we investigate this problem on perfect graphs with no balanced skew-partition; the balanced skew-partition was introduced in the decomposition theorem of Berge graphs which led to the celebrated proof of the Strong Perfect Graph Theorem. Recently, Chudnovsky, Trotignon, Trunck and Vušković proved that forbidding this unfriendly decomposition permits to recursively decompose Berge graphs (more precisely, Berge trigraphs) using 2-join and complement 2-join until reaching a “basic” graph, and in this way, they found an efficient combinatorial algorithm to color those graphs. We apply their decomposition result to prove that perfect graphs with no balanced skew-partition admit a quadratic-size Clique–Stable Set separator, by taking advantage of the good behavior of 2-join with respect to this property. We then generalize this result and prove that the Strong Erdős–Hajnal property holds in this class, which means that every such graph has a linear-size biclique or complement biclique. This is remarkable since the property does not hold for all perfect graphs (Fox, 2006), and this is motivated here by the following statement: when the Strong Erdős–Hajnal property holds in a hereditary class of graphs, then both the Erdős–Hajnal property and the polynomial Clique–Stable Set separation hold. Finally, we define the generalized k -join and generalize both our results on classes of graphs admitting such a decomposition. [ABSTRACT FROM AUTHOR]