A short cycle means a cycle of length at most 7.In this paper,we prove that planar graphs without adjacent short cycles are 3-colorable.This improves a result of Borodin et al.(2005).
WANG YingQian 1,MAO XiangHua 1,LU HuaJing 2 & WANG WeiFan 1 1 College of Mathematics,Physics and Information Engineering,Zhejiang Normal University,Jinhua 321004,China
A vertex distinguishing edge coloring of a graph G is a proper edge coloring of G such that any pair of vertices has the distinct sets of colors. The minimum number of colors required for a vertex distinguishing edge coloring of a graph G is denoted by χ s′ (G). In this paper, we obtained upper bounds on the vertex distinguishing chromatic index of 3-regular Halin graphs and Halin graphs with △(G) ≥ 4, respectively.
