General neighbour-distinguishing index via chromatic number

AbstractAn 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 gndi(G) of colours in a neighbour-distinguishing edge colouring of G. Győri et al. [E. Győri, M. Horňák, C. Palmer, M. Woźniak, General neighbour-distinguishing index of a graph, Discrete Math. 308 (2008) 827–831] proved that gndi(G)∈{2,3} 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 ⌈log2χ(G)⌉+1≤gndi(G)≤⌊log2χ(G)⌋+2. Therefore, if log2χ(G)∉Z, then gndi(G)=⌈log2χ(G)⌉+1

Another step towards proving a conjecture by Plummer and Toft

AbstractA cyclic colouring of a graph G embedded in a surface is a vertex colouring of G in which any two distinct vertices sharing a face receive distinct colours. The cyclic chromatic number χc(G) of G is the smallest number of colours in a cyclic colouring of G. Plummer and Toft in 1987 [M.D. Plummer, B. Toft, Cyclic coloration of 3-polytopes, J. Graph Theory 11 (1987) 507–515] conjectured that χc(G)≤Δ∗+2 for any 3-connected plane graph G with maximum face degree Δ∗. It is known that the conjecture holds true for Δ∗≤4 and Δ∗≥24. The validity of the conjecture is proved in the paper for Δ∗≥18

Decomposition of complete bipartite even graphs into closed trails

summary:We prove that any complete bipartite graph $K_{a,b}$, where $a,b$ are even integers, can be decomposed into closed trails with prescribed even lengths

The achromatic number of K6□K7K6 \square K7 is 18

A vertex colouring $ƒ: V(G) \rightarrow C$ of a graph $G$ is complete if for any two distinct colours $c_{1},c_{2} \in C$ there is an edge $\{v_{1} ,v_{2}\} \in E(G)$ such that $ƒ(v_{i}) = c_{i}, i = 1, 2$. The achromatic number of G is the maximum number achr(G) of colours in a proper complete vertex colouring of G. In the paper it is proved that achr$(K_{6} \square K_{7})$ = 18. This result finalises the determination of achr$(K_{6} \square K_{q})$