Vertex pancyclic graphs (original) (raw)

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.

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. ?

Pancyclic graphs and linear forests

Discrete Mathematics, 2009

Given integers k, s, t with 0 ≤ s ≤ t and k ≥ 0, a (k, t, s)-linear forest F is a graph that is the vertex disjoint union of t paths with a total of k edges and with s of the paths being single vertices. If the number of single vertex paths is not critical, the forest F will simply be called a (k, t)-linear forest. A graph G of order n ≥ k + t is (k, t)-hamiltonian if for any (k, t)-linear forest F there is a hamiltonian cycle containing F. More generally, given integers m and n with k + t ≤ m ≤ n, a graph G of order n is (k, t, s, m)-pancyclic if for any (k, t, s)-linear forest F and for each integer r with m ≤ r ≤ n, there is a cycle of length r containing the linear forest F. Minimum degree conditions and minimum sum of degree conditions of nonadjacent vertices that imply a graph is (k, t, s, m)-pancyclic (or just (k, t, m)-pancyclic) are proved.

Spectral Sufficient Conditions on Pancyclic Graphs

Complexity, 2021

A pancyclic graph of order n is a graph with cycles of all possible lengths from 3 to n . In fact, it is NP-complete that deciding whether a graph is pancyclic. Because the spectrum of graphs is convenient to be calculated, in this study, we try to use the spectral theory of graphs to study this problem and give some sufficient conditions for a graph to be pancyclic in terms of the spectral radius and the signless Laplacian spectral radius of the graph.

New sufficient conditions for hamiltonian and pancyclic graphs

Discussiones Mathematicae Graph Theory, 2007

For a graph G of order n we consider the unique partition of its vertex set V (G) = A ∪ B with A = {v ∈ V (G) : d(v) ≥ n/2} and B = {v ∈ V (G) : d(v) < n/2}. Imposing conditions on the vertices of the set B we obtain new sufficient conditions for hamiltonian and pancyclic graphs.

Minimal Degree and (k, m)-Pancyclic Ordered Graphs

Graphs and Combinatorics, 2005

Given positive integers k £ m £ n, a graph G of order n is (k, m)-pancyclic ordered if for any set of k vertices of G and any integer r with m £ r £ n, there is a cycle of length r encountering the k vertices in a specified order. Minimum degree conditions that imply a graph of sufficiently large order n is (k, m)-pancylic ordered are proved. Examples showing that these constraints are best possible are also provided.

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.