Independence, Radius and Hamiltonian Paths

Discrete Applied Mathematics, 2017

Let G be a simple connected graph with diameter d, d ̸ = 2, minimum degree δ(G) and leaf number L(G) such that δ(G) ≥ 1 2 (L(G) + 1). We show that G is traceable. The results, apart from providing a partial solution to a conjecture (Graffiti.pc 190) of the computer program Graffiti.pc, instructed by DeLaViña, provide a new sufficient condition for traceability in graphs. In addition, we provide a family of graphs to show that the results are the best and for δ(G) ≥ 9, we deduce that G is Hamiltonian.

Discrete Applied Mathematics, 1987

In this paper, an alternative closure operation to the one introduced in [2] is given. For any vertex x of a 2-connected graph G = (V, E), let N(x) be the set of vertices of V adjacent to x and d(x) be the degree of x in G. Let T= {xe. V:a, b ¢ N(x)} be a set of vertices associated with 2 nonadjacent vertices a,b and let aa~=2+ ITI and gab=minaieT d(ai). We prove that if G is 2-connected and a, b are two nonadjacent vertices such that Olab<-~gab, then G is hamiltonian if and only if G + ab is hamiltonian. Condition ~tab<_ gab is a simpler version of the main result of this paper. * This work was partly supported by the SERC, London, England. 0166-218X/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)

Information Processing Letters, 2005

A Hamiltonian cycle is a spanning cycle in a graph, i.e., a cycle through every vertex, and a Hamiltonian path is a spanning path. In this paper we present two theorems stating sufficient conditions for a graph to possess Hamiltonian cycles and Hamiltonian paths. The significance of the theorems is discussed, and it is shown that the famous Ore's theorem directly follows from our result.

Commentarii Mathematici Helvetici, 1969

The Scientific World Journal, 2014

A Hamiltonian path in a graph is a path involving all the vertices of the graph. In this paper, we revisit the famous Hamiltonian path problem and present new sufficient conditions for the existence of a Hamiltonian path in a graph.

La Matematica, 2022

Let G be a simple, connected, triangle-free graph with minimum degree δ, order n and leaf number L(G). If G has a cut-vertex, we prove that L(G) ≥ 4δ − 4 and n ≥ 4δ − 1. Both lower bounds are sharp. The lower bound on the leaf number strengthens a result by Mukwembi for triangle-free graphs. As corollaries, we deduce sufficient conditions for connectivity, traceability and Hamiltonicity in triangle-free graphs. As an easy extension of a result by Goodman and Hedetiniemi, we show that a simple, connected, claw-free, paw-free graph G is Hamiltonian if and only if G is not a path. We consider only simple graphs, that is, graphs with neither loops nor multiple edges.

Journal of Combinatorial Theory, Series B, 1981

In this paper, we derive some results giving sufficient conditions for a graph G containing a Hamiltonian path to be Hamiltonian. In particular the Bondy-Chvatal theorem [J. A. Bondy and V. Chvatal, Discrete Math. 15 (1976), 111-1351 is derived as a corollary of the main theorem of this paper and hence a more powerful closure operation than the one introduced by Bondy and Chvatal is defined. These results can be viewed as a step towards a unification of the various known results on the existence of Hamiltonian circuits in undirected graphs. Moreover, Theorem 1 of this paper provides a counterpart of the Chvatal-Erdos theorem [V. Chvatal and P. Erdiis, Discrete Math. 2 (1972), 11 l-l 131 which gives a sufficient condition for a Hamiltonian circuit in terms of global vertex connectivity and independence number.

Journal of Combinatorial Theory, Series B, 1989

Discrete Mathematics, 1993

It was claimed by that if G is a connected graph of order at least 3 such that no bridge is incident to a vertex of degree 2 and no path contains three or more consecutive vertices of degree 2, then L'(G) is hamiltonian. By H.-J. , this was proved to be false. Necessary and sufficient conditions for corresponding counterexamples were also displayed. However, this characterization is not complete because some counterexamples were overlooked as is proved in this note. Furthermore, a counterexample to a theorem of about hamiltonian index of graphs with hamiltonian cyclic blocks was exhibited by . Here an alternative version of this theorem is presented.

2009

We consider only finite undirected graphs without loops or multiple edges. Notation and terminology not defined here follow that in [2]. For a graph G = (V, E), n := |V |, e := |E|, and G := (V, E), where E := { xy : x ∈ V, y ∈ V, x = y, xy ∈ E }. For a bipartite graph GBPT = (X, Y ; E), GBPT := (X, Y ; E ), where E := { xy : x ∈ X, y ∈ Y, xy ∈ E }. The degree of vertex vi is denoted by di. The concept of closure of a general graph G was introduced by Bondy and Chvátal [1]. The k closure of a graph G, denoted clk(G), is a graph obtained from G by recursively joining two nonadjacent vertices such that their degree sum is at least k. The idea for the closure of a balanced bipartite graph can be found in [1] and [6]. The k closure of a balanced bipartite graph GBPT = (X, Y ; E), where |X | = |Y |, denoted clk(GBPT ), is a graph obtained from G by recursively joining two nonadjacent vertices x ∈ X and y ∈ Y such that their degree sum is at least k. We use C(n, r) to denote the number of...