Murty [A generalization of the Hoffman–Singleton graph, Ars Combin. 7 (1979) 191–193.] constructed a family of $(pm + 2)$-\ud
regular graphs of girth five and order $2p^{2m}$, where $p \ge 5$ is a prime, which includes the Hoffman–Singleton graph [A.J. Hoffman, R.R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. (1960) 497–504]. This construction gives an upper bound for the least number $f (k)$ of vertices of a $k$-regular graph with girth 5. In this paper, we extend the Murty construction to $k$-regular graphs with girth 5, for each $k$. In particular, we obtain new upper bounds for $f (k)$, $k \ge 16$
We study snarks whose edges cannot be covered by fewer than five perfect matchings. Esperet and Mazzuoccolo found an infinite family of such snarks, generalising an example provided by Hägglund. We construct another infinite family, arising from a generalisation in a different direction. The proof that this family has the requested property is computer-assisted. In addition, we prove that the snarks from this family (we call them treelike snarks) have circular flow number φ C (G) ≥ 5 and admit a 5-cycle double cover.
In this paper we give a method for obtaining the adjacency matrix of a simple polarity graph G q from a projective plane PG(2, q), where q is a prime power. Denote by ex(n; C 4 ) the maximum number of edges of a graph on n vertices and free of squares C 4 . We use the constructed graphs G q to obtain lower bounds on the extremal function ex(n; C 4 ), for some n < q 2 + q + 1. In particular, we construct a C 4 -free graph on n = q 2 − √ q vertices and 1 2 q(q 2 − 1) − 1 2 √ q(q − 1) edges, for a square prime power q.
A graph G is pseudo 2-factor isomorphic if the parity of the number of circuits in a 2-factor is the same for all 2-factors of G. We prove that there exist no pseudo 2-factor isomorphic k-regular bipartite graphs for k 4. We also propose a characterization for 3-edge-connected pseudo 2-factor isomorphic cubic bipartite graphs and obtain some partial results towards our conjecture.
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