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 $f (k)$, $k \ge 16$
The Heawood graph and $K_{3,3}$ have the property that all of their 2-factors are Hamilton circuits. We call such graphs 2-factor hamiltonian. We prove that if G is a k-regular bipartite 2-factor hamiltonian graph then either G is a circuit or k = 3. Furthermore, we construct an infinite family of cubic bipartite 2-factor hamiltonian graphs based on the Heawood graph and $K_{3,3}$ and conjecture that these are the only such graphs
We consider one-factorizations of K 2n possessing an automorphism group acting regularly (sharply transitively) on vertices. We present some upper bounds on the number of one-factors which are ®xed by the group; further information is obtained when equality holds in these bounds. The case where the group is dihedral is studied in some detail, with some non-existence statements in case the number of ®xed one-factors is as large as possible. Constructions both for dihedral groups and for some classes of abelian groups are given.
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.
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