Clique number of the square of a line graph

On the clique number of the square of a line graph and its relation to maximum degree of the line graph

“…In [9], Theorem 1.4 is proved by counting edges in the strong clique which are in the second neighborhood of a vertex v of maximum degree. The crucial part of the argument lies in counting the subset (called D) of edges in the strong clique which are not incident with any neighbor of v (ie, edges whose ends are both at distance exactly two from v).…”

“…In a recent article, Śleszyńska-Nowak [9] gave a new proof of Theorem 1.3. Furthermore, she improved the bound for the general case as follows.…”

A Stronger Bound for the Strong Chromatic Index

“…After our result was published on arXiv,Śleszyńska-Nowak [26] found a neat elementary proof that the size of a strong clique is always at most 3 2 Δ(G) 2 for any graph G. Coming back to strong edge colourings, let us note that there are a number of results for special graph classes. The conjecture was verified for maximum degree 3 by Andersen [2], and independently by Horek, Qing and Trotter [11].…”

Strong Cliques in Claw-Free Graphs

“…Conjecture 2 remains open, but is closer to being proved than Conjecture 1. The second author proved that x L 2 ðGÞ ð Þ 3 2 D 2 [28], which was improved to 4 3 D 2 by Faron and Postle [14]. There are also sharp results for specific classes of graphs: for bipartite graphs the upper bound is D 2 [16] (sharp for complete bipartite graphs), and for triangle-free graphs it is 5 4 D 2 [3, Theorem 4.1.5] (sharp for blowups of C 5 ).…”