Vertex pancyclic graphs

Let G be a graph of order n. A graph G is called pancyclic if it contains a cycle of length k for every 3kn, and it is called vertex pancyclic if every vertex is contained in a cycle of length k for every 3kn. In this paper, we shall present different sufficient conditions for graphs to be vertex pancyclic.     

Weakly pancyclic graphs

In generalizing the concept of a pancyclic graph, we say that a graph is “weakly pancyclic” if it contains cycles of every length between the length of a shortest and a longest cycle. In this paper it is shown that in many cases the requirements on a graph which ensure that it is weakly pancyclic are considerably weaker than those required to ensure that it is pancyclic. This sheds some light on the content of a famous metaconjecture of Bondy. From the main result of this paper it follows that 2-connected nonbipartite graphs of sufficiently large order n with minimum degree exceeding 2n/7 are weakly pancyclic; and that graphs with minimum degree at least n/4 + 250 are pancyclic, if they contain both a triangle and a hamiltonian cycle. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 141–176, 1998     

On pancyclic line graphs

In this paper, we give a best possible Ore-like condition for a graph so that its line graph is pancyclic or vertex pancyclic.     

On circuits and pancyclic line graphs

Clark proved that L(G) is hamiltonian if G is a connected graph of order n ≥ 6 such that deg u + deg v ≥ n – 1 – p(n) for every edge uv of G, where p(n) = 0 if n is even and p(n) = 1 if n is odd. Here it is shown that the bound n – 1 – p(n) can be decreased to (2n + 1)/3 if every bridge of G is incident with a vertex of degree 1, which is a necessary condition for hamiltonicity of L(G). Moreover, the conclusion that L(G) is hamiltonian can be strengthened to the conclusion that L(G) is pancyclic. Lesniak-Foster and Williamson proved that G contains a spanning closed trail if |V(G)| = n ≥ 6, δ(G) ≥ 2 and deg u + deg v ≥ n – 1 for every pair of nonadjacent vertices u and v. The bound n – 1 can be decreased to (2n + 3)/3 if G is connected and bridgeless, which is necessary for G to have a spanning closed trail.     

A note on vertex pancyclic oriented graphs

Let D be an oriented graph of order n ≥ 9, minimum degree at least n − 2, such that, for the choice of distinct vertices x and y, either xy ∈ E(D) or d⁺(x) + d⁻(y) ≥ n − 3. Song (J. Graph Theory 18 (1994), 461–468) proved that D is pancyclic. In this note, we give a short proof, based on Song's result, that D is, in fact, vertex pancyclic. This also generalizes a result of Jackson (J. Graph Theory 5 (1981), 147–157) for the existence of a hamiltonian cycle in oriented graphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 313–318, 1999     

Bondy的泛圈图定理的改进

记G=(V,E)是简单图,1971年Bondy得到O re条件下的泛圈图的著名结果:若2连通n阶图G的不相邻的任两点x、y均有d(x) d(y)≥n,则G是泛圈图或G=Kn/2,n/2.这里进一步研究条件d(x) d(y)≥n-1,得到:若2连通n阶图G的不相邻的任两点x、y均有d(x) d(y)≥n-1,则G是泛圈图或G∈{K(Cn 1)/2∨G(n-1)/2,Kn/2,n/2}.本文作者得知最近国际著名权威专家Ho lton等人也得到完全相同的结果,但本证明更简捷.     

Hamiltonian threshold graphs

The vertices of a threshold graph G are partitioned into a clique K and an independent set I so that the neighborhoods of the vertices of I are totally ordered by inclusion. The question of whether G is hamiltonian is reduced to the case that K and I have the same size, say r, in which case the edges of K do not affect the answer and may be dropped from G, yielding a bipartite graph B. Let d₁≤d₂≤…≤d_r and e₁≤e₂≤…≤e_r be the degrees in B of the vertices of I and K, respectively. For each q = 0, 1,…,r−1, denote by S_q the following system of inequalities: d_j⩾j + 1, j = 1,2,…,q, e_j⩾j + 1, j = 1, 2,…, r−1−1. Then the following conditions are equivalent:

• 1.(1) B is hamiltonian,
• 2.(2) S_q holds for some q = 0, 1,…, r−1,
• 3.(3) S_q holds for each q = 0, 1,…, r−1.

     

Hamiltonian path graphs

The Hamiltonian path graph H(G) of a graph G is that graph having the same vertex set as G and in which two vertices u and v are adjacent if and only if G contains a Hamiltonian u-v path. A characterization of Hamiltonian graphs isomorphic to their Hamiltonian path graphs is presented.     

k-ordered Hamiltonian graphs

A hamiltonian graph G of order n is k-ordered, 2 ≤ k ≤ n, if for every sequence v₁, v₂, …, v_k of k distinct vertices of G, there exists a hamiltonian cycle that encounters v₁, v₂, …, v_k in this order. Theorems by Dirac and Ore, presenting sufficient conditions for a graph to be hamiltonian, are generalized to k-ordered hamiltonian graphs. The existence of k-ordered graphs with small maximum degree is investigated; in particular, a family of 4-regular 4-ordered graphs is described. A graph G of order n ≥ 3 is k-hamiltonian-connected, 2 ≤ k ≤ n, if for every sequence v₁, v₂, …, v_k of k distinct vertices, G contains a v₁-v_k hamiltonian path that encounters v₁, v₂,…, v_k in this order. It is shown that for k ≥ 3, every (k + 1)-hamiltonian-connected graph is k-ordered and a result of Ore on hamiltonian-connected graphs is generalized to k-hamiltonian-connected graphs. © 1997 John Wiley & Sons, Inc.     

Hamiltonian line graphs

Sufficient conditions on the degrees of a graph are given in order that its line graph have a hamiltonian cycle.     

Hamiltonian graphs involving distances

Let G be a graph of order n. We show that if G is a 2-connected graph and max{d(u), d(v)} + |N(u) U N(v)| ≥ n for each pair of vertices u, v at distance two, then either G is hamiltonian or G ?3K_n/3 U T₁ U T₂, where n ? O (mod 3), and T₁ and T₂ are the edge sets of two vertex disjoint triangles containing exactly one vertex from each K_n/3. This result generalizes both Fan's and Lindquester's results as well as several others.     

Hamiltonian uniform subset graphs

The uniform subset graph G(n, k, t) is defined to have all k-subsets of an n-set as vertices and edges joining k-subsets intersecting at t elements. We conjecture that G(n, k, t) is hamiltonian when it is different from the Petersen graph and does possess cycles. We verify this conjecture for k − t = 1, 2, 3 and for suitably large n when t = 0, 1.     

On k-ordered Hamiltonian graphs

A Hamiltonian graph G of order n is k-ordered, 2 ≤ k ≤ n, if for every sequence v₁, v₂, …, v_k of k distinct vertices of G, there exists a Hamiltonian cycle that encounters v₁, v₂, …, v_k in this order. Define f(k, n) as the smallest integer m for which any graph on n vertices with minimum degree at least m is a k-ordered Hamiltonian graph. In this article, answering a question of Ng and Schultz, we determine f(k, n) if n is sufficiently large in terms of k. Let g(k, n) = − 1. More precisely, we show that f(k, n) = g(k, n) if n ≥ 11k − 3. Furthermore, we show that f(k, n) ≥ g(k, n) for any n ≥ 2k. Finally we show that f(k, n) > g(k, n) if 2k ≤ n ≤ 3k − 6. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 17–25, 1999     

On Hamiltonian bipartite graphs

Various sufficient conditions for the existence of Hamiltonian circuits in ordinary graphs are known. In this paper the analogous results for bipartite graphs are obtained.