On wreathed lexicographic products of graphs

This paper proves a necessary and sufficient condition for the endomorphism monoid of a lexicographic product G[H] of graphs G,H to be the wreath product of the monoids and . The paper also gives respective necessary and sufficient conditions for specialized cases such as for unretractive or triangle-free graphs G.     

Some relations between rank, chromatic number and energy of graphs

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. Let G be a graph of order n and be the rank of the adjacency matrix of G. In this paper we characterize all graphs with . Among other results we show that apart from a few families of graphs, , where n is the number of vertices of G, and χ(G) are the complement and the chromatic number of G, respectively. Moreover some new lower bounds for E(G) in terms of are given.     

Directed 3-cycle decompositions of complete directed graphs with quadratic leaves

Let G be a vertex-disjoint union of directed cycles in the complete directed graph D_t, let |E(G)| be the number of directed edges of G and suppose or if t=5, and if t=6. It is proved in this paper that for each positive integer t, there exist -decompositions for D_t−G if and only if .     

How to play the Majority game with a liar

The Majority game is played by a questioner () and an answerer (). holds n elements, each of which can be labeled as 0 or 1. is trying to identify some element holds as having the Majority label or, in the case of a tie, claim there is none. To do this asks questions comparing whether two elements have the same or different label. ’s goal is to ask as few questions as possible while ’s goal is to delay as much as possible. Let q^∗ denote the minimal number of questions needed for to identify a Majority element regardless of ’s answers.In this paper we investigate upper and lower bounds for q^∗ in a variation of the Majority game, where is allowed to lie up to t times. We consider two versions of the game, the adaptive (where questions are asked sequentially) and the oblivious (where questions are asked in one batch).     

On the number of graphs with a given endomorphism monoid

For a given finite monoid , let be the number of graphs on n vertices with endomorphism monoid isomorphic to . For any nontrivial monoid we prove that where and are constants depending only on with .For every k there exists a monoid of size k with , on the other hand if a group of unity of has a size k>2 then .     

Generalization of transitive fraternal augmentations for directed graphs and its applications

Let be a directed graph. A transitive fraternal augmentation of is a directed graph with the same vertex set, including all the arcs of and such that for any vertices x,y,z,

1.

if and then or (fraternity);

2.

if and then (transitivity).

In this paper, we explore some generalization of the transitive fraternal augmentations for directed graphs and its applications. In particular, we show that the 2-coloring number col₂(G)≤O(∇₁(G)∇₀(G)²), where ∇_k(G) (k≥0) denotes the greatest reduced average density with depth k of a graph G; we give a constructive proof that ∇_k(G) bounds the distance (k+1)-coloring number col_k+1(G) with a function f(∇_k(G)). On the other hand, ∇_k(G)≤(col_2k+1(G))^2k+1. We also show that an inductive generalization of transitive fraternal augmentations can be used to study nonrepetitive colorings of graphs.     

Maximum bipartite subgraphs of cubic triangle-free planar graphs

Thomassen recently proved, using the Tutte cycle technique, that if G is a 3-connected cubic triangle-free planar graph then G contains a bipartite subgraph with at least edges, improving the previously known lower bound . We extend Thomassen’s technique and further improve this lower bound to .     

Total colorings and list total colorings of planar graphs without intersecting 4-cycles

Suppose that G is a planar graph with maximum degree Δ and without intersecting 4-cycles, that is, no two cycles of length 4 have a common vertex. Let χ^″(G), and denote the total chromatic number, list edge chromatic number and list total chromatic number of G, respectively. In this paper, it is proved that χ^″(G)=Δ+1 if Δ≥7, and and if Δ(G)≥8. Furthermore, if G is a graph embedded in a surface of nonnegative characteristic, then our results also hold.     

On the signed edge domination number of graphs

Let be the signed edge domination number of G. In 2006, Xu conjectured that: for any 2-connected graph G of order n(n≥2), . In this article we show that this conjecture is not true. More precisely, we show that for any positive integer m, there exists an m-connected graph G such that . Also for every two natural numbers m and n, we determine , where K_m,n is the complete bipartite graph with part sizes m and n.     

Coloring the complements of intersection graphs of geometric figures

Let be the complement of the intersection graph G of a family of translations of a compact convex figure in Rⁿ. When n=2, we show that , where γ(G) is the size of the minimum dominating set of G. The bound on is sharp. For higher dimension we show that , for n?3. We also study the chromatic number of the complement of the intersection graph of homothetic copies of a fixed convex body in Rⁿ.     

Total domination and total domination subdivision number of a graph and its complement

A set S of vertices of a graph G=(V,E) with no isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination numberγ_t(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision numbersd_γt(G) is the minimum number of edges that must be subdivided in order to increase the total domination number. We consider graphs of order n?4, minimum degree δ and maximum degree Δ. We prove that if each component of G and has order at least 3 and , then and if each component of G and has order at least 2 and at least one component of G and has order at least 3, then . We also give a result on stronger than a conjecture by Harary and Haynes.     

Posets of annular non-crossing partitions of types B and D

We study the set of annular non-crossing permutations of type B, and we introduce a corresponding set of annular non-crossing partitions of type B, where p and q are two positive integers. We prove that the natural bijection between and is a poset isomorphism, where the partial order on is induced from the hyperoctahedral group B_p+q, while is partially ordered by reverse refinement. In the case when q=1, we prove that is a lattice with respect to reverse refinement order.We point out that an analogous development can be pursued in type D, where one gets a canonical isomorphism between and . For q=1, the poset coincides with a poset “NC^(D)(p+1)” constructed in a paper by Athanasiadis and Reiner [C.A. Athanasiadis, V. Reiner, Noncrossing partitions for the group D_n, SIAM Journal of Discrete Mathematics 18 (2004) 397-417], and is a lattice by the results of that paper.     

On (d,1)-total numbers of graphs

A (d,1)-total labelling of a graph G assigns integers to the vertices and edges of G such that adjacent vertices receive distinct labels, adjacent edges receive distinct labels, and a vertex and its incident edges receive labels that differ in absolute value by at least d. The span of a (d,1)-total labelling is the maximum difference between two labels. The (d,1)-total number, denoted , is defined to be the least span among all (d,1)-total labellings of G. We prove new upper bounds for , compute some for complete bipartite graphs K_m,n, and completely determine all for d=1,2,3. We also propose a conjecture on an upper bound for in terms of the chromatic number and the chromatic index of G.     

A note on an embedding problem in transitive tournaments

Let TT_n be a transitive tournament on n vertices. It is known Görlich, Pil?niak, Wo?niak, (2006) [3] that for any acyclic oriented graph of order n and size not greater than , two graphs isomorphic to are arc-disjoint subgraphs of TT_n. In this paper, we consider the problem of embedding of acyclic oriented graphs into their complements in transitive tournaments. We show that any acyclic oriented graph of size at most is embeddable into all its complements in TT_n. Moreover, this bound is generally the best possible.     

Large monochromatic components in colorings of complete 3-uniform hypergraphs

Let f(n,r) be the largest integer m with the following property: if the edges of the complete 3-uniform hypergraph are colored with r colors then there is a monochromatic component with at least m vertices. Here we show that and . Both results are sharp under suitable divisibility conditions (namely if n is divisible by 7, or by 6 respectively).     

Sequences of integers with missing quotients

Let S be any set of natural numbers, and A be a given set of rational numbers. We say that S is an A-quotient-free set if x,y∈S implies y/x∉A. Let and , where the supremum is taken over all A-quotient-free sets S, and are the upper and lower asymptotic densities of S respectively. Let ρ(A)=sup_Sδ(S), where the supremum is taken over all A-quotient-free sets S such that δ(S) exists. In this paper we study the properties of , and ρ(A).     

Game colouring of the square of graphs

This paper studies the game chromatic number and game colouring number of the square of graphs. In particular, we prove that if G is a forest of maximum degree Δ≥9, then , and there are forests G with . It is also proved that for an outerplanar graph G of maximum degree Δ, , and for a planar graph G of maximum degree Δ, .     

On a conjecture concerning the orientation number of a graph

For a connected graph G containing no bridges, let D(G) be the family of strong orientations of G; and for any D∈D(G), we denote by d(D) the diameter of D. The orientation number of G is defined by . Let G(p,q;m) denote the family of simple graphs obtained from the disjoint union of two complete graphs K_p and K_q by adding m edges linking them in an arbitrary manner. The study of the orientation numbers of graphs in G(p,q;m) was introduced by Koh and Ng [K.M. Koh, K.L. Ng, The orientation number of two complete graphs with linkages, Discrete Math. 295 (2005) 91-106]. Define and . In this paper we prove a conjecture on α proposed by K.M. Koh and K.L. Ng in the above mentioned paper, for q≥p+4.     

General neighbour-distinguishing index via chromatic number

An edge colouring of a graph G without isolated edges is neighbour-distinguishing if any two adjacent vertices have distinct sets consisting of colours of their incident edges. The general neighbour-distinguishing index of G is the minimum number of colours in a neighbour-distinguishing edge colouring of G. Gy?ri et al. [E. Gy?ri, M. Horňák, C. Palmer, M. Wo?niak, General neighbour-distinguishing index of a graph, Discrete Math. 308 (2008) 827-831] proved that provided G is bipartite and gave a complete characterisation of bipartite graphs according to their general neighbour-distinguishing index. The aim of this paper is to prove that if χ(G)≥3, then . Therefore, if log₂χ(G)∉Z, then .     

Facet-inducing web and antiweb inequalities for the graph coloring polytope

For a graph G and its complement , we define the graph coloring polytope P(G) to be the convex hull of the incidence vectors of star partitions of . We examine inequalities whose support graphs are webs and antiwebs appearing as induced subgraphs in G. We show that for an antiweb in G the corresponding inequality is facet-inducing for P(G) if and only if is critical with respect to vertex colorings. An analogous result is also proved for the web inequalities.