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.

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...

In the 1960’s, Tutte presented a decomposition of a 2-connected finite graph into 3-connected graphs, cycles and bonds. This decomposition has been used to reduce problems on 2-connected graphs to problems on 3-connected graphs. Motivated by a problem concerning accumulation points of infinite planar graphs, we generalize Tutte’s decomposition to include all infinite 2-connected graphs.

a fullerene graph is a 3−connected planar graph whose faces are pentagons and hexagons.the clar number of a fullerene is the maximum size of sextet patterns, the sets of disjointhexagons which are all m-alternating for a kekulé structure m of f. an exact formula of clarnumber of some fullerene graphs and a class of carbon nanocones are obtained in this paper.

We focus on the algorithm underlying the main result of [6]. This is an algebraic formula to generate all connected graphs in a recursive and efficient manner. The key feature is that each graph carries a scalar factor given by the inverse of the order of its group of automorphisms. In the present paper, we revise that algorithm on the level of graphs. Moreover, we extend the result subsequentl...

The problem on the Hamiltonicity of graphs is well studied in discrete algorithm and graph theory, because of its relation to traveling salesman problem (TSP). Starting with Tutte’s result, stating that every 4-connected planar graph is Hamiltonian, several researchers have studied the Hamiltonicity of graphs on surfaces. Extending Tutte’s technique, Thomassen proved that every 4-connected plan...