Diameter, edge-connectivity, and $C_4$-freeness

DOI10.1016/J.DISC.2023.113318arXiv2112.08805MaRDI QIDQ6385870

Vanessa Hiebeler, Johannes Pardey, Dieter Rautenbach

Publication date: 16 December 2021

Abstract: Improving a recent result of Fundikwa, Mazorodze, and Mukwembi, we show that $dleq(2n-3)/5$ for every connected $C_4$-free graph of order $n$, diameter $d$, and edge-connectivity at least $3$, which is best possible up to a small additive constant. For edge-connectivity at least $4$, we improve this to $dleq(n-3)/3$. Furthermore, adapting a construction due to ErdH{o}s, Pach, Pollack, and Tuza, for an odd prime power $q$ at least $7$, and every positive integer $k$, we show the existence of a connected $C_4$-free graph of order $n=(q^2+q-1)k+1$, diameter $d=4k$, and edge-connectivity $lambda$ at least $q-6$, in particular, $dgeq4(n-1)/(lambd{a}^2+O(lambda))$.

Mathematics Subject Classification ID

Distance in graphs (05C12) Connectivity (05C40)

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