Charles H. C. Little

A characterization of convertible (0, 1)-matrices

Abstract We characterize those (0, 1)-matrices M whose elements can be given plus or minus signs so as to yield a matrix M ′ for which det M ′ = perm M .

On defect-d matchings in graphs

A defect-d matching in a graph G is a matching which covers all but d vertices of G. G is d-covered if each edge of G belongs to a defect-d matching. Here we characterise d-covered graphs and d-covered connected bipartite graphs. We show that a regular graph G of degree r which is (r � 1)-edge-connected is 0-covered or 1-covered depending on whether G has an even or odd number of vertices, but, given any non-negative integers r and d, there exists a graph regular of degree r with connectivity and edge-connectivity r � 2 which does not even have a defect-d matching. Finally, we prove that a vertex-transitive graph is 0-covered or 1-covered depending on whether it has an even or odd number of vertices.

Cycles in bipartite tournaments

Abstract It is shown that if an oriented complete bipartite graph has a directed cycle of length 2 n , then it has directed cycles of all smaller even lengths unless n is even and the 2 n -cycle induces one special digraph.

A theorem on connected graphs in which every edge belongs to a 1-factor

We adopt the definitions and notation of [1] with the following exceptions and additions. All graphs considered are connected. Let G be a graph, and let e = {v, w} where e e E{G) and v,we V(G). It follows that G-e = G ~ {v, w}. The contradictory definition of G — e given in [1] is not used in this paper. We call v and w the ends of e. Cycles, paths and 1-factors are defined as in [1] except that they are each regarded as sets of edges rather than sequences or subgraphs. If C is a cycle, V(C) is defined as the union of the edges of C. If P is a v ~ w path, we call v and w the terminal vertices of P. The symmetric difference Ft A F2 of distinct 1-factors f x and F2 is a set of disjoint cycles; these we call alternating cycles. Thus an alternating cycle can also be defined as a cycle A of even length such that G — V(A) has a 1-factor. A graph with at least four vertices is called factor covered if for any edge there is a 1-factor that contains that edge. Now let e be an arbitrary edge of the factor covered graph G. Since G is connected and | V(G) | ^ 4 , there is another edge incident on one of the ends of e, and this edge must belong to some 1-factor F x . Therefore e

An additivity theorem for maximum genus of a graph

FY. On the other hand e belongs to some 1-factor F2 • Therefore e e Fx A F2, and e thus belongs to some alternating cycle of G. On the other hand, any edge that belongs to an alternating cycle certainly belongs to a 1-factor, and so we see that factor covered graphs are those connected graphs in which for any edge there is an alternating cycle containing that edge.

A characterization of planar cubic graphs

Abstract Let G be a finite, connected graph with no loops or multiple edges. If G is the union of two blocks, then a necessary and sufficient condition is given for the maximum genus of G to be the sum of the maximum genera of its blocks. If in addition the blocks of G are upper embeddable, then a necessary and sufficient condition is given for the upper embeddability of G.

Regular odd rings and non-planar graphs

If S is a collection of circuits in a graph G, the circuits in S are said to be consistently orientable if G can be oriented so that they are all directed circuits. If S is a set of three or more consistently orientable circuits such that no edge of G belongs to more than two circuits of S, then S is called a ring if there exists a cyclic ordering C0, C1,…, Cn − 1, C0 of the n circuits in S such that ECi ⋔ ECj ≠ ⊘ if and only if j = i or j ≡ i − 1 (mod n) or j ≡ i + 1 (mod n). We characterise planar cubic graphs in terms of the non-existence of a ring with certain specified properties.

An Extension of kasteleyn's method of enumerating the 1-factors of planar graphs

In a previous paper we have announced that a graph is non-planar if and only if it contains a maximal, strict, compact, odd ring. Little has conjectured that the compactness condition may be removed. Chernyak has now published a proof of this conjecture. However, it is difficult to test a ring for maximality. In this paper we show that for odd rings of size five or greater, the condition of maximality may be replaced by a new one called regularity. Regularity is an easier condition to diagnose than is maximality.