We consider the travelling salesman problem on metrics which can be viewed as the shortest path metric of an undirected graph with unit edge-lengths. Finding a TSP tour in such a metric is then equivalent to finding a connected Eulerian subgraph in the underlying graph. Since the length of the tour is the number of edges in this Eulerian subgraph our problem can equivalently be stated as follow...

A dual-eulerian graph is a plane graph which has an ordering defined on its edge set which forms simultaneously an Euler circuit in the graph and an euler circuit in the dual graph. Dual-eulerian graphs were defined and studied in the context of silicon optimization of cmos layouts. They are necessarily of low connectivity with exponentially many planar embeddings. We give a polynomial time alg...

d-spheres are defined graph theoretically and inductively as the empty graph in dimension d = −1 and d-dimensional graphs for which all unit spheres S(x) are (d−1)-spheres and such that for d ≥ 0 the removal of one vertex renders the graph contractible. Eulerian d-spheres are geometric d-spheres which can be colored with d + 1 colors. They are Eulerian graphs in the classical sense and for d ≥ ...

We study three mixing properties of a graph: large algebraic connectivity, large Cheeger constant (isoperimetric number) and large spectral gap from 1 for the second largest eigenvalue of the transition probability matrix of the random walk on the graph. We prove equivalence of this properties (in some sense). We give estimates for the probability for a random graph to satisfy these properties....

In this paper we prove that any distance-balanced graph $G$ with $Delta(G)geq |V(G)|-3$ is regular. Also we define notion of distance-balanced closure of a graph and we find distance-balanced closures of trees $T$ with $Delta(T)geq |V(T)|-3$.

The k-th semi total point graph of a graph G, , ‎is a graph‎ obtained from G by adding k vertices corresponding to each edge and‎ connecting them to the endpoints of edge considered‎. ‎In this paper‎, a formula for Laplacian polynomial of in terms of‎ characteristic and Laplacian polynomials of G is computed‎, ‎where is a connected regular graph‎.The Kirchhoff index of is also computed‎.

‎Let $R$ be a commutative ring with non-zero identity. ‎We describe all $C_3$‎- ‎and $C_4$-free intersection graph of non-trivial ideals of $R$ as well as $C_n$-free intersection graph when $R$ is a reduced ring. ‎Also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎Finally, ‎we shall prove that almost all Artin rings $R$ have Hamiltonian intersection graphs. ...

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.