Let G be a simple graph of order n and size m which is not a tree. If 3 is a natural number and the length of every cycle of G is divisible by , then m −2(n − 2), and the equality holds if and only if the following hold: (i) is odd and G is a cycle of order or (ii) is even and G is a generalized θ-graph with paths of length 2 . It is shown that for a (0 mod )-cycle graph, m n < −1 , if is odd, ...

Given a graph G = (V, E), V(G) = V and E(G) = E denote the vertex set and the edge set of G, respectively. All graphs considered in this paper are undirected graphs. A simple path (or path for short) is a sequence of adjacent edges (v1, v2), (v2, v3), ..., (vm-2, vm-1), (vm-1, vm), written as 〈v1, v2, v3, ..., vm〉, in which all of the vertices v1, v2, v3, ..., vm are distinct except possibly v1...

The sun is the graph obtained from a cycle of length even and at least six by adding edges to make the even-indexed vertices pairwise adjacent. Suns play an important role in the study of strongly chordal graphs. A graph is chordal if it does not contain an induced cycle of length at least four. A graph is strongly chordal if it is chordal and every even cycle has a chord joining vertices whose...

The paired de Bruijn graph is an extension of de Bruijn graph incorporating mate pair information for genome assembly proposed by Mevdedev et al. However, unlike in an ordinary de Bruijn graph, not every path or cycle in a paired de Bruijn graph will spell a string, because there is an additional soundness constraint on the path. In this paper we show that the problem of checking if there is a ...

An Archimedean graph in R 2 is any finite subgraph of the vertices and edges of one of the 11 Archimedean tilings of R E shown in Figures 1-3. Archimedean filings are commonly characterized (as in Figures 1-3) by the order in which regular polygons (with unit edges) occur around each vertex of the tiling; see Grtinbaum and Shephard [3]. We use the same description for Archimedean subgraphs. Whe...

Suppose we have some undirected graph G = (GV , GE) represented as an adjacency matrix with |GV | rows and |GV | columns. G is said to be Hamiltonian if there exists a cycle CH that passes through each vertex in GV exactly once. This cycle is called a Hamiltonian cycle. Finding a Hamiltonian cycle in a graph if one exists is an NP-complete problem. We would like to use a zeroknowledge protocol ...

The balanced Hamiltonian cycle problem is a quiet new topic of graph theorem. Given a graph G = (V, E), whose edge set can be partitioned into k dimensions, for positive integer k and a Hamiltonian cycle C on G. The set of all i-dimensional edge of C, which is a subset by E(C), is denoted as Ei(C). If ||Ei(C)| |Ej(C)|| 1 for 1 i < j k, C is called a balanced Hamiltonian cycle. In this paper, th...

Kwak, J.H. and J. Lee, Isomorphism classes of cycle permutation graphs, Discrete Mathematics 105 (1992) 131-142. In this paper, we construct a cycle permutation graph as a covering graph over the dumbbell graph, and give a new characterization of when two given cycle permutation graphs are isomorphic by a positive or a negative natural isomorphism. Also, we count the isomorphism classes of cycl...

We consider the problem of computing a minimum cycle basis in a directed graph. The input to this problem is a directed graph G whose edges have non-negative weights. A cycle in this graph is actually a cycle in the underlying undirected graph with edges traversable in both directions. A {−1,0,1} edge incidence vector is associated with each cycle: edges traversed by the cycle in the right dire...

We consider the problem of computing a minimum cycle basis in a directed graph. The input to this problem is a directed graph G whose edges have non-negative weights. A cycle in this graph is actually a cycle in the underlying undirected graph with edges traversable in both directions. A f 1;0;1g edge incidence vector is associated with each cycle: edges traversed by the cycle in the right dire...