Martin Funk

2-factor Hamiltonian graphs

The Heawood graph and K3, 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 K3,3 and conjecture that these are the only such graphs.

Graphs and digraphs with all 2-factors isomorphic

We show that a digraph which contains a directed 2-factor and has minimum in-degree and out-degree at least four has two non-isomorphic directed 2-factors. As a corollary, we deduce that every graph which contains a 2-factor and has minimum degree at least eight has two non-isomorphic 2- factors. In addition we construct: an infinite family of 3-diregular digraphs with the property that all their directed 2-factors are Hamilton cycles, an in finite family of 2-connected 4-regular graphs with the property that all their 2-factors are isomorphic, and an infinite family of cyclically 6-edge-connected cubic graphs with the property that all their 2-factors are Hamilton cycles.

A family of regular graphs of girth 5

Murty [A generalization of the Hoffman-Singleton graph, Ars Combin. 7 (1979) 191-193.] constructed a family of (p^m+2)-regular graphs of girth five and order 2p^2^m, where p>=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>=16.

Regular bipartite graphs with all 2-factors isomorphic

We prove that if G is a k-regular bipartite graph and all 2-factors of G are isomorphic then k ≤ 3.

On minimally one-factorable r -regular bipartite graphs

A corollary to the famous Marriage Theorem by Ph. Hall [7] says that every r-regular bipartite graph is one-factorable (cf. e.g. [10, Theorem 3:2]); a glance at the proof reveals that this theorem admits the following slightly sharper formulation: let G be an r-regular bipartite graph; then G has one-factors and every one-factor can be completed to a one-factorization. We will prove that the Heawood graph is an instance of a graph ful lling this theorem minimally, i.e. in which every one-factor belongs to precisely one one-factorization. When dealing with r-regular bipartite graphs G and their adjacency matrices A, useful tools are the following algebraic invariants:

Corrigendum: Corrigendum to “Graphs and digraphs with all 2-factors isomorphic” [J. Combin. Theory Ser. B 92 (2) (2004) 395--404]

We point out several errors in our article [M. Abreu, R.E.L. Aldred, M. Funk, B. Jackson, D. Labbate, J. Sheehan, Graphs and digraphs with all 2-factor isomorphic, J. Combin. Theory Ser. B 92 (2004) 395-404] which were caused by our misquoting of a theorem of C. Thomassen. We also describe how the correct statement of Thomassens theorem, together with another of his theorems, can be used to obtain weaker results than those incorrectly stated in our original article.

On configurations of type n k with constant degree of irreducibility

Abstract Symmetric schematic configurations C of type nκ correspond to n-square {0, 1}-matrices I which have κ entries 1 in each row and column, but no 2-square submatrices whose entries are all 1. With each anti-flag in C , i.e., with each entry 0 in I, we associate a weight which is defined as the number of certain 2-square submatrices of I. This paper deals with matrices I whose entries 0 have constant weight d. One has d = κ and d = 1 if, and only if, C is a projective plane of order κ − 1 and a generalized quadrangle or order (κ − 1, κ − 1), respectively. Finally, the addition and multiplication tables of GF(q) give rise to a third class of instances with parameters n = q2, κ = q, and d = κ − 1 = q − 1.