Equitable list coloring of planar graphs without 4- and 6-cycles

AbstractA graph G is equitably k-choosable if for any k-uniform list assignment L, there exists an L-colorable of G such that each color appears on at most ⌈|V(G)|k⌉ vertices. Kostochka, Pelsmajer and West introduced this notion and conjectured that G is equitably k-choosable for k>Δ(G). We prove this for planar graphs with Δ(G)≥6 and no 4- or 6-cycles

Acta Mathematica Scientia 2006,26B(3):477-482 EDGE-FACE CHROMATIC NUMBER OF 2-CONNECTED PLANE GRAPHS WITH HIGH MAXIMUM DEGREE ∗

Abstract The edge-face chromatic number χef(G) of a plane graph G is the least number of colors assigned to the edges and faces such that every adjacent or incident pair of them receives different colors. In this article, the authors prove that every 2-connected plane graph G with ∆(G) ≥ |G | − 2 ≥ 9 has χef(G) = ∆(G). Key words Plane graph, edge-face chromatic number, edge chromatic number, maxi-mum degree 2000 MR Subject Classification 05C1

Adjacent vertex distinguishing edge-colorings of planar graphs with girth at least six

An adjacent vertex distinguishing edge-coloring of a graph G is a proper edge-coloring o G such that any pair of adjacent vertices are incident to distinct sets of colors. The minimum number of colors required for an adjacent vertex distinguishing edge-coloring of G is denoted by χ'ₐ(G). We prove that χ'ₐ(G) is at most the maximum degree plus 2 if G is a planar graph without isolated edges whose girth is at least 6. This gives new evidence to a conjecture proposed in [Z. Zhang, L. Liu, and J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett., 15 (2002) 623-626.

Injective coloring of planar graphs

International audienc

On backbone coloring of graphs

International audienceLet G be a graph and H a subgraph of G. A backbone-k-coloring of (G,H) is a mapping f: V(G)→{1,2,...,k} such that |f(u)−f(v)|≥2 if uv∈E(H) and |f(u)−f(v)|≥1 if uv∈E(G)\E(H). The backbone chromatic number of (G,H) is the smallest integer k such that (G,H) has a backbone-k-coloring. In this paper, we characterize the backbone chromatic number of Halin graphs G=T∪C with respect to given spanning trees T. Also we study the backbone coloring for other special graphs such as complete graphs, wheels, graphs with small maximum average degree, graphs with maximum degree 3, etc