For many families of combinatorial objects, a construction of Wilf (1977) allows the members of the family to be viewed as paths in a directed multigraph. Introducing a partition of these paths generates a number of known, but hitherto disparate, combinatorial identities. We include several examples.

The main result of this short note is a generic version of Paz’ conjecture on the length of generating sets in matrix algebras. Consider a generic g-tuple A = (A1, . . . , Ag) of n × n matrices over an infinite field. We show that whenever g ≥ n, the set of all words of degree 2d in A spans the full n × n matrix algebra. Our proofs use generic matrices, are combinatorial and depend on the const...

In 1963, Corrádi and Hajnal proved that for all k≥1 and n≥3k, every (simple) graph G on n vertices with minimum degree δ(G)≥2k contains k disjoint cycles. The same year, Dirac described the 3-connected multigraphs not containing two disjoint cycles and asked the more general question: Which (2k− 1)-connected multigraphs do not contain k disjoint cycles? Recently, the authors characterized the s...

For each subset of edges of a (directed multi-) graph, one may determine whether the edges can be represented as a trail; for each graph, the fraction of trail-representable subsets out of the total number of subsets of edges. How large can this fraction be as a function of the number edges m? I give an upper bound of O( √ logm/ √ m) and show by example that the upper bound is nearly tight.

It is shown that a Cayley multigraph over a group G with generating multiset S is integral (i.e., all of its eigenvalues are integers) if S lies in the integral cone over the boolean algebra generated by the normal subgroups of G. The converse holds in the case when G is abelian. This in particular gives an alternative, character-theoretic proof of a theorem of Bridges and Mena (1982). We exten...

A graph G is terminal-pairable with respect to a demand multigraph D on the same vertex set as G, if there exists edge-disjoint paths joining the end vertices of every demand edge of D. In this short note, we improve the upper bound on largest ∆(n) with the property that the complete graph on n vertices is terminal-pairable with respect to any demand multigraph of maximum degree at most ∆(n) . ...

A P3-decomposition of a graph is a partition of the edges of the graph into paths of length two. We give a necessary and sufficient condition for a semi-complete multigraph, that is a multigraph with at least one edge between each pair of vertices, to have a P3-decomposition. We show that this condition can be tested in strongly polynomial-time, and that the same condition applies to a larger c...

The adaptable chromatic number of a multigraph G, denoted χa(G), is the smallest integer k such that every edge labeling of G from [k] = {1, 2, · · · , k} permits a vertex coloring of G from [k] such that no edge e = uv has c(e) = c(u) = c(v). Hell and Zhu proved that for any multigraph G with maximum degree ∆, the adaptable chromatic number is at most lp e(2∆− 1) m . We strengthen this to the ...