Chorded Pancyclicity in k-Partite Graphs (original) (raw)
Related papers
A note on pancyclicity of kkk-partite graphs
arXiv: Combinatorics, 2018
In 2009, Adamus showed that if GGG is a balanced tripartite graph of order 3n3n3n, ngeq2n \geq 2ngeq2, with at least 3n2−2n+23n^2 - 2n + 23n2−2n+2 edges, then GGG is hamiltonian and, in fact, GGG is pancyclic. Removing all but one edge incident with any vertex of the complete, balanced tripartite graph K(n,n,n)K(n,n,n)K(n,n,n) shows that this result is best possible. Here we extend the result to balanced kkk-partite graphs of order knknkn. We prove that for all integers kgeq3k\geq 3kgeq3 and ngeq1n\geq 1ngeq1, every balanced kkk-partite graph with knknkn vertices and at least (k2−k)n2−2n(k−1)+4over2{{(k^2-k)n^2-2n(k-1)+4}\over 2}(k2−k)n2−2n(k−1)+4over2 edges is pancyclic. We also prove a similar result for kkk-partite graphs that are not balanced.
Characterizations of vertex pancyclic and pancyclic ordinary complete multipartite digraphs
Discrete Mathematics, 1995
A digraph obtained by replacing each edge of a complete multipartite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a complete multipartite digraph. Such a digraph D is called ordinary if for any pair X, Y of its partite sets the set of arcs with end vertices in X ∪ Y coincides with X × Y = {x, y) : x ∈ X, y ∈ Y } or Y × X or X × Y ∪ Y × X. We characterize all the pancyclic and vertex pancyclic ordinary complete multipartite digraphs. Our characterizations admit polynomial time algorithms.
Discrete Applied Mathematics, 2002
Let G be a graph of order n. A graph G is called pancyclic if it contains a cycle of length k for every 3 6 k 6 n, and it is called vertex pancyclic if every vertex is contained in a cycle of length k for every 3 6 k 6 n. In this paper, we shall present di erent su cient conditions for graphs to be vertex pancyclic.
Some Sufficient Conditions on Pancyclic Graphs
arXiv (Cornell University), 2018
A pancyclic graph is a graph that contains cycles of all possible lengths from three up to the number of vertices in the graph. In this paper, we establish some new sufficient conditions for a graph to be pancyclic in terms of the edge number, the spectral radius and the signless Laplacian spectral radius of the graph.
Generalizing Pancyclic and k-Ordered Graphs
Graphs and Combinatorics, 2004
Given positive integers k m n, a graph G of order n is ðk; mÞ-pancyclic if for any set of k vertices of G and any integer r with m r n, there is a cycle of length r containing the k vertices. Minimum degree conditions and minimum sum of degree conditions of nonadjacent vertices that imply a graph is ðk; mÞ-pancylic are proved. If the additional property that the k vertices must appear on the cycle in a specified order is required, then the graph is said to be ðk; mÞ-pancyclic ordered. Minimum degree conditions and minimum sum of degree conditions for nonadjacent vertices that imply a graph is ðk; mÞ-pancylic ordered are also proved. Examples showing that these constraints are best possible are provided.
Hamiltonicity in balancedk-partite graphs
1995
One of the earliest results about hamiltonian graphs was given by Dirac. He showed that if a graph G has order p and minimum degree at least p then G is hamiltonian. Moon and Moser showed that a balanced bipartite graph (the two partite sets have the same order) G has order p and minimum degree more than p then G is hamiltonian. In this paper, their idea is generalized to k-partite graphs and the following result is obtained: Let G be a balanced k-partite graph with order p = kn. If the minimum degree 6(G)> i ~ k+l-n ifkisodd) 2 n k +-2 if k is even then G is hamiltonian. The result is best possible.
Pancyclicity and NP-completeness in planar graphs
Discrete Applied Mathematics, 2000
A graph is called v-pancyclic if it contains a cycle of length l containing a given vertex v for 36l6n, and a graph G is called vertex pancyclic if G is v-pancyclic for all v. In this paper, we show that it is NP-complete to determine whether a 3-connected cubic planar graph is v-pancyclic for given vertex v, it is NP-complete to determine whether a 3-connected cubic planar graph is pancyclic, and it is NP-complete to determine whether a 3-connected planar graph is vertex pancyclic. We also show that every maximal outplanar graph is vertex pancyclic. ?
Pancyclicity of 3-connected graphs: Pairs of forbidden subgraphs
Journal of Graph Theory, 2004
We characterize all pairs of connected graphs {X, Y } such that each 3-connected {X, Y }-free graph is pancyclic. In particular, we show that if each of the graphs in such a pair {X, Y } has at least four vertices, then one of them is the claw K 1,3 , while the other is a subgraph of one of six specified graphs.