Coloring Powers of Chordal Graphs

Abstract

We prove that the kth power Gk of a chordal graph G with maximum degree $\Delta$ is $O(\sqrt{k}\Delta^{(k+1)/2})$-degenerate for even values of k and $O(\Delta^{(k+1)/2})$-degenerate for odd values. In particular, this bounds the chromatic number $\chi(G^k)$ of the kth power of G. The bound proven for odd values of k is the best possible. Another consequence is the bound $\lambda_{p,q}(G)\le\lfloor\frac{(\Delta+1)^{3/2}}{\sqrt{6}}\rfloor (2q-1)+\Delta(2p-1)$ on the least possible span $\lambda_{p,q}(G)$ of an L(p,q)-labeling for chordal graphs G with maximum degree $\Delta$. On the other hand, a construction of such graphs with $\lambda_{p,q}(G)\ge\Omega(\Delta^{3/2}q+\Delta p)$ is found.