Sufficient Conditions For Hamiltonian Graphs Research Articles

Two completely independent spanning trees of split graphs

G is a split graph if V(G) can be partitioned into a clique D and an independent set I. Let τ(G) denote the toughness of a graph G. If a graph G contains two spanning trees T1,T2 such that for each two distinct vertices x,y of G, the (x,y)-path in each Ti has no common edge and no common vertex except for the two ends, then T1,T2 are called two completely independent spanning trees (CISTs) of G,i∈{1,2}. Several results have shown that some sufficient conditions for Hamiltonian graphs also guarantee the existence of two CISTs, which implies the Hamiltonicity and the existence of two CISTs of a graph have close connection. In this paper, we prove that if G is a Hamiltonian split graph with |D|>max{3,|I|}, then G contains two CISTs. Moreover, we show that if G is a Hamiltonian split graph with τ(G)>1, then G contains two CISTs. As a corollary, we obtain that every 32-tough split graph contains two CISTs.

Two completely independent spanning trees of claw-free graphs

If a graph G contains two spanning trees T1,T2 such that for each two distinct vertices x,y of G, the (x,y)-path in each Ti has no common edge and no common vertex except for the two ends, then T1,T2 are called two completely independent spanning trees (CISTs) of G,i∈{1,2}. If the subgraph induced by the neighbor set of a vertex u in a graph G is connected, then u is an eligible vertex of G. An hourglass is a graph with degree 4,2,2,2,2 and (P6)2 denotes the square graph of a path P6 on six vertices. Araki (2014) [1] proved that if a graph G satisfies the Dirac's condition, which is also sufficient condition for hamiltonian graphs, then G contains two CISTs. Ryjáček proposed that a claw-free graph G is hamiltonian if and only if its Ryjáček's closure, denoted by cl(G), is hamiltonian. In this paper, we prove that if G is a {claw, hourglass, (P6)2}-free graph with δ(G)≥4, then G contains two CISTs if and only if cl(G) has two CISTs. The bound of the minimum degree in our result is best possible.

New comments on “A Hamilton sufficient condition for completely independent spanning tree”

For k≥2, spanning trees T1,T2,…,Tk in a graph G are called to be completely independent if for any two distinct vertices x and y, the paths connecting them in T1,T2,…,Tk are pairwise openly disjoint. We call such spanning trees T1,T2,…,Tk the completely independent spanning trees (CIST for short). Recently, Hong and Zhang (2020) found that a sufficient condition for Hamiltonian graphs also suffices the existence of two CISTs. That is, if G is a graph with n vertices, |N(x)∪N(y)|≥n2 and |N(x)∩N(y)|≥3 for every two non-adjacent vertices x,y of G and n≥5, then G has two CISTs. More recently, Qin et al. (2020) proposed that the restriction on the number of vertices in the statement should be n≥8, and then pointed out that the Claim 1 in Hong’s paper is not always true for general case, which was corrected by presenting an amendment. However, there still exists a flaw in the corresponding revised proof (see details from the second part of our paper). Accordingly, we give a new amendment to correct the proof.

Comments on “A Hamilton sufficient condition for completely independent spanning tree”

Spanning trees T1,T2,…,Tk (k≥2) in a graph G are called completely independent spanning trees (CISTs for short) if for any two vertices x,y of G, the paths joining x and y in these k trees are pairwise openly disjoint. Hong and Zhang (2018) recently showed that a sufficient condition for Hamiltonian graphs still suffices for the existence of two CISTs. That is, if G is a graph with n vertices and |N(x)∪N(y)|≥n2, |N(x)∩N(y)|≥3 for every two nonadjacent vertices x,y of G and n≥5, then G admits two CISTs. In this note, we first attend that the restriction on the number of vertices in the statement should be revised. Moreover, we point out that there is a flaw in their proof. Accordingly, we give an amendment to correct the proof.

A Hamilton sufficient condition for completely independent spanning tree

Let T1,T2,…,Tk be spanning trees of a graph G. For any two vertices u,v of G, if the paths from u to v in these k trees are pairwise openly disjoint, then we say that T1,T2,…,Tk are completely independent. It has been asked whether sufficient conditions for Hamiltonian graphs are also sufficient for the existence of two completely independent spanning trees (CISTs). In this paper, we prove that it is true for aHamilton-condition. That is, if G is a graph with n vertices and |N(x)∪N(y)|≥n2, |N(x)∩N(y)|≥3 for every two nonadjacent vertices x,y of G, then G has two completely independent spanning trees.

Sufficient conditions for Hamiltonian graphs in terms of (signless Laplacian) spectral radius

Let G be a simple graph with n vertices, m edges and minimum degree . In this paper, we prove that ifthen G is Hamiltonian with only a few exceptions. Using this result, we present some spectral sufficient conditions for a graph to be Hamiltonian in terms of the spectral radius or signless Laplacian spectral radius of the graph. Our results improve some previous work.

Equivalent Condition and Two Algorithms for Hamiltonian Graphs

In this paper we present a necessary and sufficient condition for Hamiltonian graphs and also twoalgorithms and two examples in another part.

Ore’s condition for completely independent spanning trees

Two edge-disjoint spanning trees of a graph G are completely independent if the two paths connecting any two vertices of G in the two trees are internally disjoint. It has been asked whether sufficient conditions for hamiltonian graphs are also sufficient for the existence of two completely independent spanning trees (CISTs). We prove that it is true for the classical Ore-condition. That is, if G is a graph on n vertices in which each pair of non-adjacent vertices have degree-sum at least n, then G has two CISTs. It is known that the line graph of every 4-edge connected graph is hamiltonian. We prove that this is also true for CISTs: the line graph of every 4-edge connected graph has two CISTs. Thomassen conjectured that every 4-connected line graph is hamiltonian. Unfortunately, being 4-connected is not enough for the existence of two CISTs in line graphs. We prove that there are infinitely many 4-connected line graphs that do not have two CISTs.

Dirac's Condition for Completely Independent Spanning Trees

AbstractTwo spanning trees T1 and T2 of a graph G are completely independent if, for any two vertices u and v, the paths from u to v in T1 and T2 are internally disjoint. In this article, we show two sufficient conditions for the existence of completely independent spanning trees. First, we show that a graph of n vertices has two completely independent spanning trees if the minimum degree of the graph is at least . Then, we prove that the square of a 2‐connected graph has two completely independent spanning trees. These conditions are known to be sufficient conditions for Hamiltonian graphs.

One sufficient condition for Hamiltonian graphs involving distances

In 1990 G. T. Chen proved that if G is a 2-connected graph of order n and 2|N(x) ∪ N(y)| + d(x) + d(y) ≥ 2n − 1 for each pair of nonadjacent vertices x, y ∈ V (G), then G is Hamiltonian. In this paper we prove that if G is a 2-connected graph of order n and 2|N(x) ∪ N(y)| + d(x)+d(y) ≥ 2n−1 for each pair of nonadjacent vertices x, y ∈ V (G) such that d(x, y) = 2, then G is Hamiltonian.

Variations on a sufficient condition for Hamiltonian graphs

Variations on a sufficient condition for Hamiltonian graphs

A new sufficient condition for Hamiltonian graphs

The study of Hamiltonian graphs began with Dirac’s classic result in 1952. This was followed by that of Ore in 1960. In 1984 Fan generalized both these results with the following result: If G is a 2-connected graph of order n and max{d(u), d(v)}≥n/2 for each pair of vertices u and v with distance d(u, v)=2, then G is Hamiltonian. In 1991 Faudree–Gould–Jacobson–Lesnick proved that if G is a 2-connected graph and |N(u)∪N(v)|+δ(G)≥n for each pair of nonadjacent vertices u, v∈V(G), then G is Hamiltonian. This paper generalizes the above results when G is 3-connected. We show that if G is a 3-connected graph of order n and max{|N(x)∪N(y)|+d(u),|N(w)∪N(z)|+d(v)}≥n for every choice of vertices x, y, u, w, z, v such that d(x, y)=d(y, u)=d(w, z)=d(z, v)=d(u, v)=2 and where x, y and u are three distinct vertices and w, z and v are also three distinct vertices (and possibly |{x, y}∩{w, z}| is 1 or 2), then G is Hamiltonian.

New sufficient condition for Hamiltonian graphs

Let G be a graph, and δ ( G ) and α ( G ) be the minimum degree and the independence number of G , respectively. For a vertex v ∈ V ( G ) , d ( v ) and N ( v ) represent the degree of v and the neighborhood of v in G , respectively. A number of sufficient conditions for a connected simple graph G of order n to be Hamiltonian have been proved. Among them are the well known Dirac condition (1952) ( δ ( G ) ≥ n 2 ) and Ore condition (1960) (for any pair of independent vertices u and v , d ( u ) + d ( v ) ≥ n ). In 1984 Fan generalized these two conditions and proved that if G is a 2-connected graph of order n and max { d ( x ) , d ( y ) } ≥ n / 2 for each pair of nonadjacent vertices x , y with distance 2 in G , then G is Hamiltonian. In 1993, Chen proved that if G is a 2-connected graph of order n , and if max { d ( x ) , d ( y ) } ≥ n / 2 for each pair of nonadjacent vertices x , y with 1 ≤ | N ( x ) ∩ N ( y ) | ≤ α ( G ) − 1 , then G is Hamiltonian. In 1996, Chen, Egawa, Liu and Saito further showed that if G is a k -connected graph of order n , and if max { d ( v ) : v ∈ S } ≥ n / 2 for every independent set S of G with | S | = k which has two distinct vertices x , y ∈ S such that the distance between x and y is 2, then G is Hamiltonian. In this paper, we generalize all the above conditions and prove that if G is a k -connected graph of order n , and if max { d ( v ) : v ∈ S } ≥ n / 2 for every independent set S of G with | S | = k which has two distinct vertices x , y ∈ S satisfying 1 ≤ | N ( x ) ∩ N ( y ) | ≤ α ( G ) − 1 , then G is Hamiltonian.

Extension of several sufficient conditions for Hamiltonian graphs

Extension of several sufficient conditions for Hamiltonian graphs

A generalization of Bondy's and Fan's sufficient conditions for Hamiltonian graphs

Let G be a k-connected ( k ⩾ 2) graph on n vertices. Let S be an independent set of G. S is called essential if there exists two distinct vertices in S which have a common neighbor in G. In this paper we shall prove that if max { d( u) : u ∈ S} ⩾ n/2 holds for any essential independent set S with k + 1 vertices of G, then either G is hamiltonian or G is one of three classes of exceptional graphs. This is motivated by a result of Chen et al. (1994) and generalizes the results of Bondy (1980) and Fan (1984).

Old Hamiltonian Ideas from a New Point of View

We prove a sufficient condition for graphs to be hamiltonian. This result generalizes five sufficient conditions for hamiltonian graphs and is non-comparable with many well-known ones.

A common generalization of Chvátal-Erdős' and Fraisse's sufficient conditions for hamiltonian graphs

Let G = ( V, E) be a k-connected graph of order n. For S ⊂ V, let N( S) be its neighborhood set and let J( S) = { u ∉ S | N( u) ⊇ S} if | S| ⩾ 2 and J( S) = 0 otherwise. If there exists some s, 1 ⩽ s ⩽ k, such that every independent set X of s + 1 vertices has a vertex u satisfying | N( Xβ{ u})| + | N( u) ∪ J( X) β{ u})| ⩾ n, then G is hamiltonian. From this main theorem, we derive new sufficient conditions for hamiltonian graphs. Some of these results are improvements and/or generalizations of various known results. In particular, sufficient conditions of Ore (1960), Chvátal and Erdős (1972), Fraisse (1986) and E. Flandrin et al. (1991) are easily derived.

Computation of the 0-dual closure for hamiltonian graphs

The well-known closure concept of Bondy and Chvátal (1976) is based on degree sums of pairs of nonadjacent vertices. It generalizes six earlier sufficient degree conditions for hamiltonian graphs. Ainouche and Christofides (1987) introduced a more general concept which is called the 0-dual closure. We prove that this concept generalizes five sufficient conditions for hamiltonian graphs which are not generalized by the concept of Bondy and Chvátal (1976).

An improvement of fraisse's sufficient condition for hamiltonian graphs

AbstractLet G be a k‐connected graph of order n. For an independent set c, let d(S) be the number of vertices adjacent to at least one vertex of S and > let i(S) be the number of vertices adjacent to at least |S| vertices of S. We prove that if there exists some s, 1 ≤ s ≤ k, such that ΣxiEX d(X\{Xi}) > s(n−1) – k[s/2] – i(X)[(s−1)/2] holds for every independetn set X ={x0, x1 ⃛xs} of s + 1 vertices, then G is hamiltonian. Several known results, including Fraisse's sufficient condition for hamiltonian graphs, are dervied as corollaries.

Four sufficient conditions for hamiltonian graphs

The purpose of this communication is to describe four new sufficient conditions for the existence of hamiltonian circuits in graphs. We mention first ten known results. Each one can be derived from at least one of the new conditions. We consider only undirected graphs G=( V, E) without loops. We assume that G is k-connected. For S ⊂ N, we denote by N( S) the set of vertices adjacent to at least one vertex of S and by d( S) the cardinality of this set.