We present an algorithm that finds, for each vertex of an undirected graph, a shortest cycle containing it. While for directed graphs this problem reduces to the All-Pairs Shortest Paths problem, this is not known to be the case for undirected graphs. We present a truly sub-cubic randomized algorithm for the undirected case. Given an undirected graph with n vertices and integer weights in 1, . ...

Let ~m = {0, 1 . . . . , m 1}. A n m-cycle is a graph (v0, V l , . . . , u~_ 1) with vertex set {viii ~ E m} and edge set {{vi, Vi+a}]i ~ 7/m} (reducing the subscript modulo m). A n m-cycle system of a graph G is an o rdered pair (V, C), where V is the vertex set of G and C is a set of m-cycles, the edges of which part i t ion the edges of G. A n m-cycle system (of order n) is an m-cycle system...

Abstract The graph-parameters mainly studies vertex degree and distance between unordered vertices. There are many graph-parameters, this paper focuses on the Gutman index. For sake of discussion, puts forward concept “ contribution” based relationship index graph vertex-Gutman On basis, we discussed unicyclic with pendent edges in odd even parts, gave corresponding extremal graphs, which lays ...

The reload cost refers to the cost that occurs along a path on an edge-colored graph when it traverses an internal vertex between two edges of different colors. Galbiati et al. [1] introduced the Minimum Reload Cost Cycle Cover problem, which is to find a set of vertex-disjoint cycles spanning all vertices with minimum reload cost. They proved that this problem is strongly NP-hard and not appro...

A cycle in a graph G is called a hamilton cycle if it contains every vertex of G. A l-factor of a graph G is a subgraph H of G with the same vertex set as G, such that each vertex of H has degree one. Ringel [S] has generalized the idea of a hamilton cycle to two dimensions. He showed that if n is odd the set of squares in the n-dimensional cube Q,, can be partitioned into subsets such that eac...

For any integer $k\geq 3$ , we define sunlet graph of order $2k$, denoted by $L_{2k}$, as the consisting a cycle length $k$ together with pendant vertices, each adjacent to exactly one vertex cycle. In this paper, give necessary and sufficient conditions for existence $L_{8}$-decomposition tensor product wreath complete graphs.

It is shown that the maximal number of edges in an n-vertex graph containing no cycle of length six has order of magnitude 12n 4/3.

Presentation of set matrices and demonstration of their efficiency as a tool using the path/cycle problem. Introduction Set matrices are matrices whose elements are sets. The matrices comprise abilities of data storing and processing. That makes them a promising combinatorial structure. To prove the concept, this work applies the matrices to the path/cycle problem, see [1, 2, 3, 4, 5, 6, 7, 9, ...

Given a vertex-weighted graph G = (V,E) and a set S ⊆ V , a subset feedback vertex set X is a set of the vertices of G such that the graph induced by V \ X has no cycle containing a vertex of S. The Subset Feedback Vertex Set problem takes as input G and S and asks for the subset feedback vertex set of minimum total weight. In contrast to the classical Feedback Vertex Set problem which is obtai...

Thomassen (J. Combin. Theory Ser. B 28, 1980, 142–163) proved that every strong tournament contains a vertex x such that each arc going out from x is contained in a Hamiltonian cycle. In this paper, we extend the result of Thomassen and prove that a strong tournament contains a vertex x such that every arc going out from x is pancyclic, and our proof yields a polynomial algorithm to nd such a v...