We show that an Eulerian triangulation of the Klein bottle has chromatic number equal to six if and only if it contains a complete graph of order six, and it is 5-colorable, otherwise. As a consequence of our proof, we derive that every Eulerian triangulation of the Klein bottle with facewidth at least four is 5-colorable.

In this paper we obtain the expectation and variance of the number of Euler tours of a random Eulerian directed graph with fixed out-degree sequence. We use this to obtain the asymptotic distribution of the number of Euler tours of a random d-in/d-out graph and prove a concentration result. We are then able to show that a very simple approach for uniform sampling or approximately counting Euler...

For a graph G, the irregularity and total irregularity of G are defined as irr(G)=∑_(uv∈E(G))〖|d_G (u)-d_G (v)|〗 and irr_t (G)=1/2 ∑_(u,v∈V(G))〖|d_G (u)-d_G (v)|〗, respectively, where d_G (u) is the degree of vertex u. In this paper, we characterize all ‎connected Eulerian graphs with the second minimum irregularity, the second and third minimum total irregularity value, respectively.

Algebraic techniques are used to find several new combinatorial interpretations for valuations of the Martin polynomial, M(G; s), for unoriented graphs. The Martin polynomial of a graph, introduced by Martin in his 1977 thesis, encodes information about the families of closed paths in Eulerian graphs. The new results here are found by showing that the Martin polynomial is a translation of a uni...

A graph is supereulerian if it has a spanning Eulerian subgraph. Motivated by the Chinese Postman Problem, Boesch, Suffel, and Tindell ([2]) in 1997 proposed the supereulerian problem, which seeks a characterization of graphs that have spanning Eulerian subgraphs, and they indicated that this problem would be very difficult. Pulleyblank ([71]) later in 1979 proved that determining whether a gra...

De Bruijn graphs provide a framework for genome assembly, where the correct reconstruction of the genome is given by one of the many Eulerian tours through the graph. The assembly problem is complicated by genomic repeats, which allow for many possible Eulerian tours, thereby increasing the de Bruijn graph complexity. Optical maps provide an ordered listing of restriction fragment sizes for a g...

The Stochastic Eulerian Tour Problem (SETP) seeks the Eulerian tour of minimum expected length on an undirected Eulerian graph, when demand on the arcs that have to be serviced is probabilistic. In an earlier paper, we have shown that the SETP is NP-hard and derived several properties of the optimal tour. In this paper, we develop three constructive heuristics for the SETP. The first two are gr...

Bounds for the chromatic number and for some related parameters of a graph are obtained by applying algebraic techniques. In particular, the following result is proved: If G is a directed graph with maximum outdegree d, and if the number of Eulerian subgraphs of G with an even number of edges differs from the number of Eulerian subgraphs with an odd number of edges then for any assignment of a ...

For a graph G = (V,E) and a vertex v ∈ V , let T (v) be a local trace at v, i.e. T (v) is an Eulerian subgraph of G such that every walk W (v), with start vertex v can be extended to an Eulerian tour in T (v). We prove that every maximum edge-disjoint cycle packing Z∗ of G induces a maximum trace T (v) at v for every v ∈ V . Moreover, if G is Eulerian then sufficient conditions are given that g...