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, .     

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.     

Edge colouring by total labellings

We introduce the concept of an edge-colouring total k-labelling. This is a labelling of the vertices and the edges of a graph G with labels 1,2,…,k such that the weights of the edges define a proper edge colouring of G. Here the weight of an edge is the sum of its label and the labels of its two endvertices. We define to be the smallest integer k for which G has an edge-colouring total k-labelling. This parameter has natural upper and lower bounds in terms of the maximum degree Δ of . We improve the upper bound by 1 for every graph and prove . Moreover, we investigate some special classes of graphs.     

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.     

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 .     

Ore-type conditions implying 2-factors consisting of short cycles

For every graph G, let . The main result of the paper says that every n-vertex graph G with contains each spanning subgraph H all whose components are isomorphic to graphs in . This generalizes the earlier results of Justesen, Enomoto, and Wang, and is a step towards an Ore-type analogue of the Bollobás-Eldridge-Catlin Conjecture.     

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 .     

On signed cycle domination in graphs

Let G=(V,E) be a graph, a function f:E→{−1,1} is said to be an signed cycle dominating function (SCDF) of G if ∑_e∈E(C)f(e)≥1 holds for any induced cycle C of G. The signed cycle domination number of G is defined as is an SCDF of G}. In this paper, we obtain bounds on , characterize all connected graphs G with , and determine the exact value of for some special classes of graphs G. In addition, we pose some open problems and conjectures.     

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.     

On the circumference of a graph and its complement

Let G be a graph of order n and circumference c(G). Let be the complement of G. We prove that and show sharpness of this bound.     

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.     

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 .     

Disjoint hamiltonian cycles in bipartite graphs

Let G=(X,Y) be a bipartite graph and define . Moon and Moser [J. Moon, L. Moser, On Hamiltonian bipartite graphs, Israel J. Math. 1 (1963) 163-165. MR 28 # 4540] showed that if G is a bipartite graph on 2n vertices such that , then G is hamiltonian, sharpening a classical result of Ore [O. Ore, A note on Hamilton circuits, Amer. Math. Monthly 67 (1960) 55] for bipartite graphs. Here we prove that if G is a bipartite graph on 2n vertices such that , then G contains k edge-disjoint hamiltonian cycles. This extends the result of Moon and Moser and a result of R. Faudree et al. [R. Faudree, C. Rousseau, R. Schelp, Edge-disjoint Hamiltonian cycles, Graph Theory Appl. Algorithms Comput. Sci. (1984) 231-249].     

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.     

Cubicity, boxicity, and vertex cover

A k-dimensional box is the cartesian product R₁×R₂×?×R_k where each R_i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G is the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the cartesian product R₁×R₂×?×R_k where each R_i is a closed interval on the real line of the form [a_i,a_i+1]. The cubicity of G, denoted as cub(G), is the minimum k such that G is the intersection graph of a collection of k-cubes. In this paper we show that cub(G)≤t+⌈log(n−t)⌉−1 and , where t is the cardinality of a minimum vertex cover of G and n is the number of vertices of G. We also show the tightness of these upper bounds.F.S. Roberts in his pioneering paper on boxicity and cubicity had shown that for a graph G, and , where n is the number of vertices of G, and these bounds are tight. We show that if G is a bipartite graph then and this bound is tight. We also show that if G is a bipartite graph then . We point out that there exist graphs of very high boxicity but with very low chromatic number. For example there exist bipartite (i.e., 2 colorable) graphs with boxicity equal to . Interestingly, if boxicity is very close to , then chromatic number also has to be very high. In particular, we show that if , s≥0, then , where χ(G) is the chromatic number of G.     

Acyclic edge colouring of planar graphs without short cycles

Let G=(V,E) be any finite graph. A mapping C:E→[k] is called an acyclic edgek-colouring of G, if any two adjacent edges have different colours and there are no bichromatic cycles in G. In other words, for every pair of distinct colours i and j, the subgraph induced in G by all the edges which have colour i or j, is acyclic. The smallest number k of colours, such that G has an acyclic edge k-colouring is called the acyclic chromatic index of G, denoted by .In 2001, Alon et al. conjectured that for any graph G it holds that ; here Δ(G) stands for the maximum degree of G.In this paper we prove this conjecture for planar graphs with girth at least 5 and for planar graphs not containing cycles of length 4,6,8 and 9. We also show that if G is planar with girth at least 6. Moreover, we find an upper bound for the acyclic chromatic index of planar graphs without cycles of length 4. Namely, we prove that if G is such a graph, then .     

Inverse degree and edge-connectivity

Let G be a connected graph with vertex set V(G), order n=|V(G)|, minimum degree δ and edge-connectivity λ. Define the inverse degree of G as , where d(v) denotes the degree of the vertex v. We show that if

     

The hamiltonian index of a 2-connected graph

Let G be a graph. Then the hamiltonian index h(G) of G is the smallest number of iterations of line graph operator that yield a hamiltonian graph. In this paper we show that for every 2-connected simple graph G that is not isomorphic to the graph obtained from a dipole with three parallel edges by replacing every edge by a path of length l≥3. We also show that for any two 2-connected nonhamiltonian graphs G and with at least 74 vertices. The upper bounds are all sharp.     

Homomorphisms from sparse graphs to the Petersen graph

Let G be a graph and let c: be an assignment of 2-elements subsets of the set {1,…,5} to the vertices of G such that for any two adjacent vertices u and v,c(u) and c(v) are disjoint. Call such a coloring c a (5, 2)-coloring of G. A graph is (5,2)-colorable if and only if it has a homomorphism to the Petersen graph.The maximum average degree of G is defined as . In this paper, we prove that every triangle-free graph with is homomorphic to the Petersen graph. In other words, such a graph is (5, 2)-colorable. Moreover, we show that the bound on the maximum average degree in our result is best possible.     

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 .