Let G be a 2-connected claw-free graph. We show that • if G is N1,1,4-free or N1,2,2-free or Z5-free or P8-free, respectively, then G has a spanning eulerian subgraph (i.e. a spanning connected even subgraph) or its closure is the line graph of a graph in a family of well-defined graphs, • if the minimum degree δ(G) ≥ 3 and G is N2,2,5-free or Z9-free, respectively, then G has a spanning euleri...

In this paper, we consider the asymptotic properties of the numbers of spanning trees and Eulerian trials in circulant digraphs and graphs. Let C(p, s1, s2, . . . , sk) be a directed or undirected circulant graph. Let T (C(p, s1, s2, . . . , sk)) and E(C(p, s1, s2, . . . , sk)) be the numbers ∗This work was partially supported by the Natural Science Foundation of China, email: fjzhang@jingxian....

Identities obtained by elementary finite Fourier analysis are used to derive a variety of evaluations of the Tutte polynomial of a graph G on the hyperbolae H 2 and H 4. These evaluations are expressed in terms of eulerian subgraphs of G and the size of subgraphs modulo 2, 3, 4 or 6. In particular, a graph is found to have a nowhere-zero 4-flow if and only if there is a correlation between the ...

The Zagreb indices are among the oldest and the most famous topological molecular structure-descriptors. The first Zagreb index is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index is equal to the sum of the products of the degrees of pairs of adjacent vertices of the respective graph. In this paper, we characterize the extremal graphs with maximal, sec...

Given two nonnegative integers s and t , a graph G is (s, t)-supereulerian if for any disjoint sets X, Y ⊂ E(G) with |X | ≤ s and |Y | ≤ t , there is a spanning eulerian subgraph H of G that contains X and avoids Y . We prove that if G is connected and locally k-edge-connected, thenG is (s, t)-supereulerian, for any pair of nonnegative integers s and t with s+t ≤ k−1. We further show that if s ...

This paper describes the method to construct a maze for which the solution path goes along with the line of a closed one stroke drawn curve. If the crossing of the curve is considered a vertex and the curve between the two crossings (vertices) an edge, the closed curve becomes an Eulerian graph. Since maze does not allow the solution path to intersect itself, the proposed method constructs the ...

It has been shown [Balister, 2001] that if n is odd and m1, . . . , mt are integers with mi ≥ 3 and ∑t i=1 mi = |E(Kn)| then Kn can be decomposed as an edge-disjoint union of closed trails of lengths m1, . . . , mt. This result was later generalized [Balister, to appear] to all sufficiently dense Eulerian graphs G in place of Kn. In this article we consider the corresponding questions for direc...

Suppose G is a graph embedded in S g with width (also known as edge width) at least 264(2 g ?1). If P V (G) is such that the distance between any two vertices in P is at least 16, then any 5-coloring of P extends to a 5-coloring of all of G. We present similar extension theorems for 6-and 7-chromatic toroidal graphs, for 3-colorable large-width graphs embedded on S g with every face even sided,...

The goal of genome assembly is to reconstruct contiguous portions of a genome (known as contigs) given short reads of DNA sequence obtained in a sequencing experiment. De Bruijn graphs are constructed by finding overlaps of length k − 1 between all substrings of length k from the reads, resulting in a graph where the correct reconstruction of the genome is given by one of the many possible Eule...