Elsevier

European Journal of Combinatorics

Volume 42, November 2014, Pages 26-48
European Journal of Combinatorics

Improper coloring of sparse graphs with a given girth, I: (0,1)-colorings of triangle-free graphs

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Abstract

A graph G is (0,1)-colorable if V(G) can be partitioned into two sets V0 and V1 so that G[V0] is an independent set and G[V1] has maximum degree at most 1. The problem of verifying whether a graph is (0,1)-colorable is NP-complete even in the class of planar graphs of girth 9.

The maximum average degree, mad(G), of a graph G is the maximum of 2|E(H)||V(H)| over all subgraphs H of G. It was proved recently that every graph G with mad(G)125 is (0,1)-colorable, and this is sharp. This yields that every planar graph with girth at least 12 is (0,1)-colorable.

Let F(g) denote the supremum of a such that for some constant bg every graph G with girth g and |E(H)|a|V(H)|+bg for every HG is (0,1)-colorable. By the above, F(3)=1.2. We find the exact value for F(4) and F(5): F(4)=F(5)=119. In fact, we also find the best possible values of b4 and b5: every triangle-free graph G with |E(H)|<11|V(H)|+59 for every HG is (0,1)-colorable, and there are infinitely many not (0,1)-colorable graphs G with girth 5, |E(G)|=11|V(G)|+59 and |E(H)|<11|V(H)|+59 for every proper subgraph H of G. A corollary of our result is that every planar graph of girth 11 is (0,1)-colorable. This answers a half of a question by Dorbec, Kaiser, Montassier and Raspaud. In a companion paper, we show that for every g, F(g)1.25 and resolve some similar problems for the so-called (j,k)-colorings generalizing (0,1)-colorings.

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