A strong edge-colouring of a graph is a proper edge-colouring where each colour class induces a matching. It is known that every planar graph with maximum degree∆ has a strong edge-colouring with at most 4∆ + 4 colours. We show that 3∆ + 1 colours suffice if the graph has girth 6, and 4∆ colours suffice if ∆ ≥ 7 or the girth is at least 5. In the last part of the paper, we raise some questions ...

Homomorphisms between graphs are studied as a generalization of colorings and of chromatic number. We investigate here homomorphisms from orienta-tions of undirected planar graphs to graphs (not necessarily planar) containing as few digons as possible. We relate the existence of such homomorphisms to girth and it appears that these questions remain interesting even under large girth assumption ...

In this article we propose new methods for computing the asymptotic value for the logarithm of the partition function (free energy) for certain statistical physics models on certain type of finite graphs, as the size of the underlying graph goes to infinity. The two models considered are the hard-core (independent set) model when the activity parameter λ is small, and also the Potts (q-coloring...

We describe an algorithm for generating all k-critical H-free graphs, based on a method of Hoàng et al. Using this algorithm, we prove that there are only finitely many 4-critical (P7, Ck)-free graphs, for both k = 4 and k = 5. We also show that there are only finitely many 4-critical graphs (P8, C4)free graphs. For each case of these cases we also give the complete lists of critical graphs and...

A natural digraph analogue of the graph-theoretic concept of an ‘independent set’ is that of an ‘acyclic set’, namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets. In the spirit of a famous theorem of P. Erdős [Graph theory and probability, Canad. J. Math. 11 (1959), 34–38], it was shown prob...

We describe an algorithm for generating all k-critical H-free graphs, based on a method of Hoàng et al. Using this algorithm, we prove that there are only finitely many 4-critical (P7, Ck)-free graphs, for both k = 4 and k = 5. We also show that there are only finitely many 4-critical (P8, C4)-free graphs. For each case of these cases we also give the complete lists of critical graphs and verte...

In a graph G of maximum degree 3, let γ(G) denote the largest fraction of edges that can be 3 edge-coloured. Rizzi [9] showed that γ(G) ≥ 1 − 2 3godd(G) where godd(G) is the odd girth of G, when G is triangle-free. In [3] we extended that result to graph with maximum degree 3. We show here that γ(G) ≥ 1− 2 3godd(G)+2 , which leads to γ(G) ≥ 15 17 when considering graphs with odd girth at least ...

We denote by χg(G) the game chromatic number of a graph G, which is the smallest number of colors Alice needs to win the coloring game on G. We know from Montassier et al. [M. Montassier, P. Ossona de Mendez, A. Raspaud and X. Zhu, Decomposing a graph into forests, J. Graph Theory Ser. B, 102(1):38-52, 2012] and, independantly, from Wang and Zhang, [Y. Wang and Q. Zhang. Decomposing a planar gr...

We introduce some constructions of weakly distance-regular digraphs of girth 2, and prove that a certain quotient digraph of a commutative weakly distance-transitive digraph of girth 2 is a distancetransitive graph. As an application of the result, we obtain some examples of weakly distanceregular digraphs which are not weakly distance-transitive. Moreover, a class of commutative weakly distanc...

An injective coloring of a graph is a vertex coloring where two vertices have distinct colors if a path of length two exists between them. In this paper some results on injective colorings of planar graphs with few colors are presented. We show that all planar graphs of girth ≥19 and maximum degree ∆ are injectively ∆-colorable. We also show that all planar graphs of girth ≥10 are injectively (...