An improved bound for the strong clique index of graphs

Abstract

For a graph $G$ with line graph $L(G)$, $\chi (L(G{)}^2)$ and $\omega (L(G{)}^2)$ are called the \emph{strong chromatic index} and \emph{strong clique index} of $G$, respectively. A well-known conjecture of Erdős and Nešetřil (1985) posits that $\chi (L(G{)}^2)\le \frac{5}{4}\Delta (G{)}^2$. Related to that, Faudree, Gyárfás, Schelp and Tuza (1990) conjectured that $\omega (L(G{)}^2)\le \frac{5}{4}\Delta (G{)}^2$. We show that $\omega (L(G{)}^2)\le \frac{2607}{1987}\Delta (G{)}^2<\frac{21}{16}\Delta (G{)}^2$ improving the upper bound $\frac{4}{3}\Delta (G{)}^2$ of Faron and Postle. Indeed, we make progress towards a stronger conjecture of Faron and Postle in terms of Ore-degree. For positive integers $\Delta$ and $t$, let $h_t(\Delta )$ denote the smallest integer such that any graph $G$ with size at least $h_t(\Delta )$ and maximum degree $\Delta (G)\le \Delta$, contains two edges with distance at least $t$. An old problem of Erdős and Nešetřil (1986) concerns estimating the quantity $h_t(\Delta )$ and can be thought of as the edge-version of the degree-diameter problem. Chung, Gyárfás, Tuza and Trotter established the sharp inequality $h_2(\Delta )\le \frac{5}{4}{\Delta }^2+1$. We disprove two conjectures of Cambie, Cames van Batenburg, Joannis de Verclos and Kang concerning the next open case $h_3(\Delta )$.