## 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 of**G*, 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 \(\chi _{2}(G) \ge \varOmega (k^2/\log (k))\).

## Keywords

Graph ranking Star coloring## Notes

### Acknowledgements

This research was supported in part by NSA Grant H98230-14-1-0325.

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