Grinberg’s theorem is a necessary condition for the planar Hamilton graphs. In this paper, we use cycle bases and removable cycles to survey cycle structures of the Hamiltonian graphs and derive an equation of the interior faces in Grinberg’s Theorem. The result shows that Grinberg’s Theorem is suitable for the connected and simple graphs. Furthermore, by adding a new constraint of solutions to...

Abstract. A directed graph (digraph) is formed by vertices and arcs (directed edges) from one vertex to another. A feedback vertex set (FVS) is a set of vertices that contains at least one vertex of every directed cycle in this digraph. The directed feedback vertex set problem aims at constructing a FVS of minimum cardinality. This is a fundamental cycle-constrained hard combinatorial optimizat...

Assume that m, n and s are integers with m ≥ 2, n ≥ 4, 0 ≤ s < n and s is of the same parity of m. The generalized honeycomb tori GHT(m, n, s) have been recognized as an attractive architecture to existing torus interconnection networks in parallel and distributed applications. A bipartite graphG is bipancyclic if it contains a cycle of every even length from 4 to |V (G)| inclusive. G is vertex...

Using a recent result of Chudnovsky, Seymour, and Sullivan, we slightly improve two bounds related to the Caccetta-Haggkvist Conjecture. Namely, we show that if α ≥ 0.35312, then each n-vertex digraph D with minimum outdegree at least αn has a directed 3-cycle. If β ≥ 0.34564, then every n-vertex digraph D in which the outdegree and the indegree of each vertex is at least βn has a directed 3-cy...

This paper proposes a method for computing the expectation for the length of the maximum set of vertex-disjoint cycles in a digraph where vertices and/or arcs are subject to failure with a known probability. This method has an immediate practical application: it can be used for the solution of a kidney exchange program in the common situation where the underlying graph is unreliable. Results fo...

A cycle in a graph is dominating if every vertex lies at distance at most one from the cycle and a cycle is D-cycle if every edge is incident with a vertex of the cycle. In this paper, first we provide recursive formulae for finding a shortest dominating cycle in a Hahn graph; minor modifications can give formulae for finding a shortest D-cycle. Then, dominating cycles and D-cycles in a Halin g...

Let G be a graph of order n. A graph G is called pancyclic if it contains a cycle of length k for every 36 k6 n, and it is called vertex pancyclic if every vertex is contained in a cycle of length k for every 36 k6 n. In this paper, we shall present di0erent su1cient conditions for graphs to be vertex pancyclic. ? 2002 Elsevier Science B.V. All rights reserved.

A digraph is a symmetric cycle if it is symmetric and its underlying graph is a cycle. It is proved that if D is an asymmetric digraph not containing a symmetric cycle, then D remains asymmetric after removing some vertex. It is also showed that each digraph D without a symmetric cycle, whose underlying graph is connected, contains a vertex which is a common fixed point of all automorphisms of D.

In this note we show that deciding the existence of a Hamiltonian cycle in a cubic plane graph is equivalent to the problem of the existence of an associated cubic plane multi-3-gonal graph with a Hamiltonian cycle which takes alternately left and right edges at each successive vertex, i.e. it is also a Petrie cycle. The Petrie Hamiltonian cycle in an n-vertex plane cubic graph can be recognize...

For a graph G, let fvs and cp denote the minimum size of a feedback vertex set in G and the maximum size of a cycle packing in G, respectively. Kloks, Lee, and Liu conjectured that fvs(G) ≤ 2 cp(G) if G is planar. They proved a weaker inequality, replacing 2 by 5. We improve this, replacing 5 by 3, and verifying the resulting inequality for graphs embedded in surfaces of nonnegative Euler chara...