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

Abstract

A graph G is (0, 1)-colorable if V (G) can be partitioned into two sets V-0 and V-1 so that G[V-0] is an independent set and G[V-1] has maximum degree at most I. 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 vertical bar E(H)vertical bar/vertical bar V(H)vertical bar over all subgraphs H of G. It was proved recently that every graph G with mad(G) <= 12/5 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 b(g) every graph G with girth g and vertical bar E(H)vertical bar <= a vertical bar V(H)vertical bar b(g) for every H subset of G 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) = 11/9. In fact, we also find the best possible values of b(4) and b(5): every triangle-free graph G with vertical bar E(H)vertical bar < 11 vertical bar V(H)vertical bar+5/9 for every H subset of G is (0, 1)-colorable, and there are infinitely many not (0, 1)-colorable graphs G with girth 5, vertical bar E(G)vertical bar = 11 vertical bar V(G)vertical bar+5/9 and vertical bar E(H)vertical bar < 11 vertical bar V(H)vertical bar+5/9 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. (C) 2014 Elsevier Ltd. All rights reserved.