Let D be a digraph and let λ(D) be the arc-strong connectivity of D, and α′(D) be the size of a maximum matching of D. We proved that if λ(D) ≥ α′(D) > 0, then D has a spanning eulerian subdigraph. C © 2015 Wiley Periodicals, Inc. J. Graph Theory 81: 393–402, 2016

A cellular embedding of an Eulerian digraph D into a closed surface is said to be directed if the boundary of each face is a directed closed walk in D. The directed genus polynomial of an Eulerian digraph D is the polynomial ΓD(x) = ∑ h≥0 gh(D)x h where gh(D) is the number of directed embeddings into the orientable surface Sh, of genus h, for h = 0, 1, . . . . The sequence {gh(D)|h ≥ 0}, which ...

Let G be a connected graph with maximum degree ∆. Brooks’ theorem states that G has a ∆-coloring unless G is a complete graph or an odd cycle. A graph G is degree-choosable if G can be properly colored from its lists whenever each vertex v gets a list of d(v) colors. In the context of list coloring, Brooks’ theorem can be strengthened to the following. Every connected graph G is degree-choosabl...

Part of this paper summarizes some of the recent developments in the study of hamiltonian line graphs and the related hamiltonian claw-free graphs. The last section of this paper solves some problems on the hamiltonian like indices from a paper by Clark and Wormald in 1983. 1. Definitions and Terminology Graphs considered here are finite and loopless. Unless otherwise noted, we follow [2] for n...

In this paper the concept of pathos adjacency cut vertex jump graph PJC(T ) of a tree T is introduced. We also present a characterization of graphs whose pathos adjacency cut vertex jump graphs are planar, outerplanar, minimally non-outerplanar, Eulerian and Hamiltonian.

Let be the following strategy to construct a walk in a labeled digraph: at each vertex, we follow the unvisited arc of minimum label. In this work we study for which languages, applying the previous strategy over the corresponding de Bruijn graph, we finish with an Eulerian cycle, in order to obtain the minimal de Bruijn sequence of the language.

This paper describes a method for affine-invariant syntactic pattern recognition of geometric patterns using a new type of context-free graph grammar. The grammar accommodates variability in the geometric relations between parts of patterns; this variability is modelled using affine transformations and metric tensors. A parallel parsing algorithm is outlined, which is suitable for non-Eulerian ...

Multimatroids are combinatorial structures that generalize matroids and arise in the study of Eulerian graphs. We prove, by means of an efficient algorithm, a covering theorem for multimatroids. This theorem extends Edmonds’ covering theorem for matroids. It also generalizes a theorem of Jackson on the Euler tours of a 4-regular graph.

The purpose of this paper is to obtain a necessary and sufficient condition for the tensor product of two or more graphs to be connected, bipartite or eulerian. Also, we present a characterization of the duplicate graph G⊕K2 to be unicyclic. Finally, the girth and the formula for computing the number of triangles in the tensor product of graphs are worked out.

We prove the following Turán-Type result: If there are more than 9mn/16 edges in a simple and bipartite Eulerian digraph with vertex partition size m and n, then the graph contains a directed cycle of length 4 or 6. By using this result, we improve an upper bound for the diameter of interchange graphs.