Graph 2-rankings

Abstract

A 2-ranking of a graph G is an ordered partition of the vertices of G into independent sets $$V_1, \ldots , V_t$$ such that for $$i<j$$ , the subgraph of G induced by $$V_i \cup V_j$$ is a star forest in which each vertex in $$V_i$$ has degree at most 1. A 2-ranking is intermediate in strength between a star coloring and a distance-2 coloring. The 2-ranking number ofG, denoted $$\chi _{2}(G)$$ , is the minimum number of parts needed for a 2-ranking. For the d-dimensional cube $$Q_d$$ , we prove that $$\chi _{2}(Q_d) = d+1$$ . As a corollary, we improve the upper bound on the star chromatic number of products of cycles when each cycle has length divisible by 4. Let $$\chi _{2}'(G)=\chi _{2}(L(G))$$ , where L(G) is the line graph of G; equivalently, $$\chi _{2}'(G)$$ is the minimum t such that there is an ordered partition of E(G) into t matchings $$M_1, \ldots , M_t$$ such that for each j, the matching $$M_j$$ is induced in the subgraph of G with edge set $$M_1 \cup \cdots \cup M_j$$ . We show that $$\chi _{2}'(K_{m,n})=nH_m$$ when m! divides n, where $$K_{m,n}$$ is the complete bipartite graph with parts of sizes m and n, and $$H_m$$ is the harmonic sum $$1 + \cdots + \frac{1}{m}$$ . We also prove that $$\chi _{2}(G) \le 7$$ when G is subcubic and show the existence of a graph G with maximum degree k and $$\chi _{2}(G) \ge \varOmega (k^2/\log (k))$$ .