Generalized arboricity of graphs with large girth

Abstract

The arboricity of a graph G is the least number of colors needed to color the edges of G so that no cycle is monochromatic. We consider a higher order analog of this parameter, denoted by arbp(G), introduced recently by Nešetřil et al. (2014). It is defined as the least number of colors needed to color the edges of a graph so that each cycle C gets at least min{|C|,p+1} colors. So, arb1(G) is the usual arboricity of G, while arb2(G) can be seen as a relaxed version of the acyclic chromatic index of G. By the results of Nešetřil et al. (2014) it follows that arbp(G) is bounded for classes of graphs with bounded expansion, provided that the girth of graphs G is sufficiently large (depending on p). Using more direct approach we obtain explicit upper bounds on arbp(G) for some classes of graphs. In particular, we prove that arbp(G)≤p+1 for every planar graph of girth at least 2p+1. A similar result holds for graphs with arbitrary fixed genus. We also demonstrate that arb2(G)≤5 for outerplanar graphs, which is best possible. By using entropy compression argument we prove that arbp(G)≤(Δ−1)p+1 for graphs of maximum degree Δ and sufficiently large girth.