Spanning subgraphs with specified valencies

Spanning subgraphs with specified valencies

The author has published a necessary and sufficient condition for a fmite loopless graph to have a spanning subgraph with a specified positive valency at each vertex (see [8,9). In the present paper it is contended that the condition can be made more useful as a tool of graph theory by imposing a maximality condition. 1. The condition for an I-factor Let G be a finite graph. Loops and multiple ...

Enumeration of Spanning Subgraphs with Degree Constraints

For a finite undirected multigraph G = (V, E) and functions f, g : V → N, let N f (G, j) denote the number of (f, g)– factors of G with exactly j edges. The Heilmann-Lieb Theorem implies that ∑ j N 1 0 (G, j)t is a polynomial with only real (negative) zeros, and hence that the sequence N 0 (G, j) is strictly logarithmically concave. Separate generalizations of this theorem were obtained by Ruel...

Two-connected Spanning Subgraphs with at Most

On a Spanning Tree with Specified Leaves

Let k ≥ 2 be an integer. We show that if G is a (k + 1)-connected graph and each pair of nonadjacent vertices in G has degree sum at least |G| + 1, then for each subset S of V (G) with |S| = k, G has a spanning tree such that S is the set of endvertices. This result generalizes Ore’s theorem which guarantees the existence of a Hamilton path connecting any two vertices.

Reconfiguring spanning and induced subgraphs

Subgraph reconfiguration is a family of problems focusing on the reachability of the solution space in which feasible solutions are subgraphs, represented either as sets of vertices or sets of edges, satisfying a prescribed graph structure property. Although there has been previous work that can be categorized as subgraph reconfiguration, most of the related results appear under the name of the...