%A Śleszyńska-Nowak, Małgorzata
%A Dębski, Michał
%D 2019
%T t-Strong cliques and the degree-diameter problem
%K
%X 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.
%U http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/1258
%J Acta Mathematica Universitatis Comenianae
%0 Journal Article
%P 1057-1061%V 88
%N 3
%@ 0862-9544
%8 2019-07-31