Degree conditions for k‐ordered hamiltonian graphs

For a positive integer k, a graph G is k-ordered hamiltonian if for every ordered sequence of k vertices there is a hamiltonian cycle that encounters the vertices of the sequence in the given order. It is shown that if G is a graph of order n with 3 ≤ k ≤ n/2, and deg(u) + deg(v) ≥ n + (3k − 9)/2 for every pair u, v of nonadjacent vertices of G, then G is k-ordered hamiltonian. Minimum degree conditions are also given for k-ordered hamiltonicity. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 199–210, 2003     

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.     

Hamiltonicity of regular 2-connected graphs

Let G be a k-regular 2-connected graph of order n. Jackson proved that G is hamiltonian if n ≤ 3k. Zhu and Li showed that the upper bound 3k on n can be relaxed to 22/7k if G is 3-connected and k ≥ 63. We improve both results by showing that G is hamiltonian if n ≤ 7/2k − 7 and G does not belong to a restricted class F of nonhamiltonian graphs of connectivity 2. To establish this result we obtain a variation of Woodall's Hopping Lemma and use it to prove that if n ≤ 7/2k − 7 and G has a dominating cycle (i.e., a cycle such that the vertices off the cycle constitute an independent set), then G is hamiltonian. We also prove that if n ≤ 4k − 3 and G ∉ F, then G has a dominating cycle. For k ≥ 4 it is conjectured that G is hamiltonian if n ≤ 4k and G ∉ F. © 1996 John Wiley & Sons, Inc.     

Mutually Independent Hamiltonian Connectivity of (n, k)-Star Graphs

A graph G is hamiltonian connected if there exists a hamiltonian path joining any two distinct nodes of G. Two hamiltonian paths and of G from u to v are independent if u = u ₁ = v ₁, v = u _v(G) = v _v(G) , and u _i ≠ v _i for every 1 < i < v(G). A set of hamiltonian paths, {P ₁, P ₂, . . . , P _k }, of G from u to v are mutually independent if any two different hamiltonian paths are independent from u to v. A graph is k mutually independent hamiltonian connected if for any two distinct nodes u and v, there are k mutually independent hamiltonian paths from u to v. The mutually independent hamiltonian connectivity of a graph G, IHP(G), is the maximum integer k such that G is k mutually independent hamiltonian connected. Let n and k be any two distinct positive integers with n–k ≥ 2. We use S _n,k to denote the (n, k)-star graph. In this paper, we prove that IHP(S _n,k ) = n–2 except for S _4,2 such that IHP(S _4,2) = 1.      

Claw‐free graphs and 2‐factors that separate independent vertices

In this article, we prove that a line graph with minimum degree δ≥7 has a spanning subgraph in which every component is a clique of order at least three. This implies that if G is a line graph with δ≥7, then for any independent set S there is a 2‐factor of G such that each cycle contains at most one vertex of S. This supports the conjecture that δ≥5 is sufficient to imply the existence of such a 2‐factor in the larger class of claw‐free graphs. It is also shown that if G is a claw‐free graph of order n and independence number α with δ≥2n/α?2 and n≥3α³/2, then for any maximum independent set S, G has a 2‐factor with α cycles such that each cycle contains one vertex of S. This is in support of a conjecture that δ≥n/α≥5 is sufficient to imply the existence of a 2‐factor with α cycles, each containing one vertex of a maximum independent set. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 251–263, 2012     

Long cycles passing through a specified path in a graph

For a graph G and an integer k ≥ 1, let ς_k(G) = d_G(v_i): {v₁, …, v_k} is an independent set of vertices in G}. Enomoto proved the following theorem. Let s ≥ 1 and let G be a (s + 2)-connected graph. Then G has a cycle of length ≥ min{|V(G)|, ς₂(G) − s} passing through any path of length s. We generalize this result as follows. Let k ≥ 3 and s ≥ 1 and let G be a (k + s − 1)-connected graph. Then G has a cycle of length ≥ min{|V(G)|, − s} passing through any path of length s. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 177–184, 1998     

Packing in regular graphs

A set S of vertices in a graph G is a packing if the vertices in S are pairwise at distance at least 3 apart in G. The packing number of G, denoted by ρ(G), is the maximum cardinality of a packing in G. Favaron [Discrete Math. 158 (1996), 287–293] showed that if G is a connected cubic graph of order n different from the Petersen graph, then ρ(G) ≥ n/8. In this paper, we generalize Favaron’s result. We show that for k ≥ 3, if G is a connected k-regular graph of order n that is not a diameter-2 Moore graph, then ρ(G) ≥ n/(k² ? 1).     

Essential independent sets and long cycles

An independent set S of a graph G is said to be essential if S has a pair of vertices that are distance two apart in G. For SV(G) with S≠, let Δ(S)=max{d_G(x)|xS}. We prove the following theorem. Let k2 and let G be a k-connected graph. Suppose that Δ(S)d for every essential independent set S of order k. Then G has a cycle of length at least min{|G|,2d}. This generalizes a result of Fan.     

Cycles containing all vertices of maximum degree

For a graph G and an integer k, denote by V_k the set {v ∈ V(G) | d(v) ≥ k}. Veldman proved that if G is a 2-connected graph of order n with n ≤ 3k - 2 and |V_k| ≤ k, then G has a cycle containing all vertices of V_k. It is shown that the upper bound k on |V_k| is close to best possible in general. For the special case k = δ(G), it is conjectured that the condition |V_k| ≤ k can be omitted. Using a variation of Woodall's Hopping Lemma, the conjecture is proved under the additional condition that n ≤ 2δ(G) + δ(G) + 1. This result is an almost-generalization of Jackson's Theorem that every 2-connected k-regular graph of order n with n ≤ 3k is hamiltonian. An alternative proof of an extension of Jackson's Theorem is also presented. © 1993 John Wiley & Sons, Inc.     

On spanning connected graphs

A k-containerC(u,v) of G between u and v is a set of k internally disjoint paths between u and v. A k-container C(u,v) of G is a k^*-container if the set of the vertices of all the paths in C(u,v) contains all the vertices of G. A graph G is k^*-connected if there exists a k^*-container between any two distinct vertices. Therefore, a graph is 1^*-connected (respectively, 2^*-connected) if and only if it is hamiltonian connected (respectively, hamiltonian). In this paper, a classical theorem of Ore, providing sufficient conditional for a graph to be hamiltonian (respectively, hamiltonian connected), is generalized to k^*-connected graphs.     

A characterization of panconnected graphs satisfying a local ore-type condition

It is well known that a graph G of order p ≥ 3 is Hamilton-connected if d(u) + d(v) ≥ p + 1 for each pair of nonadjacent vertices u and v. In this paper we consider connected graphs G of order at least 3 for which d(u) + d(v) ≥ |N(u) ∪ N(v) ∪ N(w)| + 1 for any path uwv with uv ∉ E(G), where N(x) denote the neighborhood of a vertex x. We prove that a graph G satisfying this condition has the following properties: (a) For each pair of nonadjacent vertices x, y of G and for each integer k, d(x, y) ≤ k ≤ |V(G)| − 1, there is an x − y path of length k. (b) For each edge xy of G and for each integer k (excepting maybe one k η {3,4}) there is a cycle of length k containing xy. Consequently G is panconnected (and also edge pancyclic) if and only if each edge of G belongs to a triangle and a quadrangle. Our results imply some results of Williamson, Faudree, and Schelp. © 1996 John Wiley & Sons, Inc.     

Edge-dominating cycles in graphs

A set S of vertices in a graph G is said to be an edge-dominating set if every edge in G is incident with a vertex in S. A cycle in G is said to be a dominating cycle if its vertex set is an edge-dominating set. Nash-Williams [Edge-disjoint hamiltonian circuits in graphs with vertices of large valency, Studies in Pure Mathematics, Academic Press, London, 1971, pp. 157-183] has proved that every longest cycle in a 2-connected graph of order n and minimum degree at least is a dominating cycle. In this paper, we prove that for a prescribed positive integer k, under the same minimum degree condition, if n is sufficiently large and if we take k disjoint cycles so that they contain as many vertices as possible, then these cycles form an edge-dominating set. Nash-Williams’ Theorem corresponds to the case of k=1 of this result.     

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.     

Degree-bounded coloring of graphs: Variations on a theme by brooks

A graph G is degree-bounded-colorable (briefly, db-colorable) if it can be properly vertex-colored with colors 1,2, …, k ≤ Δ(G) such that each vertex v is assigned a color c(v) ≤ v. We first prove that if a connected graph G has a block which is neither a complete graph nor an odd cycle, then G is db-colorable. One may think of this as an improvement of Brooks' theorem in which the global bound Δ(G) on the number of colors is replaced by the local bound deg v on the color at vertex v. Extending the above result, we provide an algorithmic characterization of db-colorable graphs, as well as a nonalgorithmic characterization of db-colorable trees. We briefly examine the problem of determining the smallest integer k such that G is db-colorable with colors 1, 2,…, k. Finally, we extend these results to set coloring. © 1995, John Wiley & Sons, Inc.     

On 3-regular 4-ordered graphs

A simple graph G is k-ordered (respectively, k-ordered hamiltonian), if for any sequence of k distinct vertices v₁,…,v_kof G there exists a cycle (respectively, hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability and posed the question of the existence of 3-regular 4-ordered (hamiltonian) graphs other than K₄ and K_3,3. Ng and Schultz observed that a 3-regular 4-ordered graph on more than 4 vertices is triangle free. We prove that a 3-regular 4-ordered graph G on more than 6 vertices is square free,and we show that the smallest graph that is triangle and square free, namely the Petersen graph, is 4-ordered. Furthermore, we prove that the smallest graph after K₄ and K_3,3 that is 3-regular 4-ordered hamiltonianis the Heawood graph. Finally, we construct an infinite family of 3-regular 4-ordered graphs.     

Hamiltonian cycles in n‐extendable graphs

A graph G of order at least 2n+2 is said to be n‐extendable if G has a perfect matching and every set of n independent edges extends to a perfect matching in G. We prove that every pair of nonadjacent vertices x and y in a connected n‐extendable graph of order p satisfy deg_G x+deg_G y ≥ p ? n ? 1, then either G is hamiltonian or G is isomorphic to one of two exceptional graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 75–82, 2002     

Partitioning a graph into convex sets

Let G be a finite simple graph. Let S⊆V(G), its closed interval I[S] is the set of all vertices lying on shortest paths between any pair of vertices of S. The set S is convex if I[S]=S. In this work we define the concept of a convex partition of graphs. If there exists a partition of V(G) into p convex sets we say that G is p-convex. We prove that it is NP-complete to decide whether a graph G is p-convex for a fixed integer p≥2. We show that every connected chordal graph is p-convex, for 1≤p≤n. We also establish conditions on n and k to decide if the k-th power of a cycle C_n is p-convex. Finally, we develop a linear-time algorithm to decide if a cograph is p-convex.     

Matching extension in the powers of n-connected graphs

Let G be a graph on p vertices. Then for a positive integer n₁ G is said to be n-extendible if (i) n < p/2, (ii) G has a set of n independent edges, and (iii) every such set is contained in a perfect matching of G. In this paper we will show that if p is even and G is n-connected, then G^k is -extendible for every integer k ≥ 2 such that . © 1996 John Wiley & Sons, Inc.     

The entire graph of a bridgeless connected plane graph is hamiltonian

In this paper we show that the entire graph of a bridgeless connected plane graph is hamiltonian, and that the entire graph of a plane block is hamiltonian connected and vertex pancyclic. In addition, we show that in any block G which is not a circuit, given a vertex v of G and a circuit k of G, there is a path p, suspended in G, such that p is a path in k of length at least 1 and G ? E(p) ? V₀(G ? E(p)) is a block which includes v.     

Relative length of long paths and cycles in graphs with large degree sums

For a graph G, p(G) denotes the order of a longest path in G and c(G) the order of a longest cycle. We show that if G is a connected graph n ≥ 3 vertices such that d(u) + d(v) + d(w) ≧ n for all triples u, v, w of independent vertices, then G satisfies c(G) ≥ p(G) – 1, or G is in one of six families of exceptional graphs. This generalizes results of Bondy and of Bauer, Morgana, Schmeichel, and Veldman. © 1995, John Wiley & Sons, Inc.