t-Strong cliques and the degree-diameter problem

Abstract

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 \emph{$t$-strong chromatic index} of $G$ is the chromatic number of $L(G)^t$ and a \emph{$t$-strong clique} in $G$ is a clique in $L(G)^t$. Finding upper bounds for the \emph{$t$-strong chromatic index} and \emph{$t$-strong clique} are problems related to two famous problems: the conjecture of Erd{\H o}s and Ne{\v s}et{\v r}il 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 $\Delta^t+O\left(\Delta^{t-1}\right)$. We also show results for some special classes of graphs: $K_{1,r}$-free graphs and graphs with a large girth.