Backbone coloring for graphs with large girths

For a graph G and a subgraph H (called backbone graph) of G, a backbone k-coloring of G with respect to H is a proper vertex coloring of G using colors from the set {1, 2, . . . , k}, with an additional condition that colors for any two adjacent vertices in H must differ by at least two. The backbone chromatic number of G over H, denoted by BBC(G,H), is the smallest k of a backbone k-coloring admitted by G with respect to H. Broersma, Fomin, Golovach, and Woeginger [2] showed that BBC(G,H) ≤ 2χ(G) − 1 holds for every G and H; moreover, for every n there exists a graph G with a spanning tree T such that χ(G) = n and the bound is sharp. To answer a question raised in [2], Mǐskuf, Škrekovski, and Tancer [17] proved that for any n there exists a triangle-free graph G with a spanning tree T such that χ(G) = n and BBC(G,T ) = 2n − 1. We extend this result by showing that for any positive integers n and l, there exists a graph G with a spanning tree T such that G has girth at least l, χ(G) = n, and BBC(G,T ) = 2n− 1. ∗Department of Mathematics, Zhejiang Normal University, China. Grant Number: NSFC No. 11271334 and ZJNSF No. Z6110786. Email: [email protected] †Department of Mathematics, California State University, Los Angeles, Email: [email protected] ‡Department of Mathematics, Zhejiang Normal University, China. Grant Number: NSFC No. 11171310 and ZJNSF No. Z6110786. Email: [email protected]