t-STRONG CLIQUES AND THE DEGREE-DIAMETER PROBLEM.

For a graph $G$, $L(G)^t$ is the $t$th power of the line graph of $G$; that is, vertices of $L(G)^t$ are edges of $G$ and two edges $e,f\in E(G)$ are adjacent in $L(G)^t$ if $G$ contains a path with at most $t$ vertices that starts in a vertex of $e$ and ends in a vertex of $f$. The distance-$t$ chromatic index of $G$ is the chromatic number of $L(G)^t$, and a $t$-strong clique in $G$ is a clique in $L(G)^t$. Finding upper bounds for the distance-$t$ chromatic index and $t$-strong clique are problems related to two famous problems: the conjecture of Erdös and Nešetřil concerning the strong chromatic index, and the degree/diameter problem. We prove that the size of a $t$-strong clique in a graph with maximum degree $\Delta$ is at most $1.75\Delta^t+O\left(\Delta^{t-1}\right)$, and for bipartite graphs the upper bound is at most $\Delta^t+O\left(\Delta^{t-1}\right)$. As a corollary, we obtain upper bounds of $1.881\Delta^t +O\left(\Delta^{t-1}\right)$ and $1.9703+O\left(\Delta^{t-1}\right)$ on the distance-$t$ chromatic index of bipartite graphs and general graphs. We also show results for some special classes of graphs: $K_{1,r}$-free graphs and graphs with a large girth. [ABSTRACT FROM AUTHOR]