We show that the edges of any graph $G$ containing two edge-disjoint spanning trees can be blue/red coloured so blue and red graphs are connected degrees at each vertex differ by most four. This improves a result H\"orsch. discuss variations question for digraphs, infinite computational question, resolve further questions H\"orsch in negative.

Let G = (V, E) be an oriented graph whose edges are labelled by the elements of a group Γ. A cycle C in G has non-zero weight if for a given orientation of the cycle, when we add the labels of the forward directed edges and subtract the labels of the reverse directed edges, the total is non-zero. We are specifically interested in the maximum number of vertex disjoint non-zero cycles. We prove t...

A graph G is weakly 4-connected if it is 3-connected, has at least five vertices, and for every pair (A,B) such that A ∪ B = V (G), |A ∩ B| = 3 and no edge has one end in A − B and the other in B − A, one of the induced subgraphs G[A], G[B] has at most four edges. We describe a set of constructions that starting from a weakly 4-connected planar graph G produce a finite list of non-planar weakly...

Let (G) be the domination number of graph G, thus a graph G is k -edge-critical if (G) 1⁄4 k ; and for every nonadjacent pair of vertices u and v, (Gþ uv) 1⁄4 k 1. In Chapter 16 of the book ‘‘Domination in Graphs— Advanced Topics,’’ D. Sumner cites a conjecture of E. Wojcicka under the form ‘‘3-connected 4-critical graphs are Hamiltonian and perhaps, in general (i.e., for any k 4), (k 1)-connec...

A non-crossing connected graph is a connected graph on vertices arranged in a circle such that its edges do not cross. The count for such graphs can be made naturally into a q-binomial generating function. We prove that this generating function exhibits the cyclic sieving phenomenon, as conjectured by S.-P. Eu. Résumé. Un graphe connexe dont les sommets sont disposés sur un cercle est sans croi...

It is well known that 3–regular graphs with arbitrarily large girth exist. Three constructions are given that use the former to produce non-Hamiltonian 3–regular graphs without reducing the girth, thereby proving that such graphs with arbitrarily large girth also exist. The resulting graphs can be 1–, 2– or 3–edge-connected depending on the construction chosen. From the constructions arise (nai...

Yutsis graphs are connected simple graphs which can be partitioned into two vertex-induced trees. Cubic Yutsis graphs were introduced by Jaeger as cubic dual Hamiltonian graphs, and these are our main focus. Cubic Yutsis graphs also appear in the context of the quantum theory of angular momenta, where they are used to generate summation formulae for general recoupling coefficients. Large Yutsis...

Let A be a minor-closed class of labelled graphs, and let Gn be a random graph sampled uniformly from the set of n-vertex graphs of A. When n is large, what is the probability that Gn is connected? How many components does it have? How large is its biggest component? Thanks to the work of McDiarmid and his collaborators, these questions are now solved when all excluded minors are 2-connected. U...

In 1966 Barnette introduced a set of graphs, called circuit graphs, which are obtained from 3-connected planar graphs by deleting a vertex. Circuit graphs and 3-connected planar graphs share many interesting properties which are not satisfied by general 2-connected planar graphs. Circuit graphs have nice closure properties which make them easier to deal with than 3-connected planar graphs for s...