Hamilton cycles in split graphs with large minimum degree

Discrete Mathematics, 2002

Let G be a graph with n vertices and minimum degree at least n=2, and B a set of vertices with at least 3n=4 vertices. In this paper, we show that there exists a hamiltonian cycle in which every vertex in B is adjacent to some vertex in B.

Discrete Mathematics, 2009

Let G = (X, Y ) be a bipartite graph and define σ 2 2 (G) = min{d(x) + d(y) : xy / ∈ E(G), x ∈ X, y ∈ Y }. Moon and Moser showed that if G is a bipartite graph on 2n vertices such that σ 2 2 (G) ≥ n + 1 then G is hamiltonian, sharpening a classical result of Ore [6] for bipartite graphs. Here we prove that if G is a bipartite graph on 2n vertices such that σ 2 2 (G) ≥ n + 2k − 1 then G contains k edge-disjoint hamiltonain cycles. This extends the result of Moon and Moser and a result of R. Faudree, et al.

Cornell University - arXiv, 2017

In this paper we investigate Hamiltonian path problem in the context of split graphs, and produce a dichotomy result on the complexity of the problem. Our main result is a deep investigation of the structure of K1,4-free split graphs in the context of Hamiltonian path problem, and as a consequence, we obtain a polynomial-time algorithm to the Hamiltonian path problem in K1,4-free split graphs. We close this paper with the hardness result: we show that, unless P=NP, Hamiltonian path problem is NP-complete in K1,5-free split graphs by reducing from Hamiltonian cycle problem in K1,5-free split graphs. Thus this paper establishes a "thin complexity line" separating NP-complete instances and polynomial-time solvable instances. Lemma 3 (Chvatal[21]). Let G be a connected graph. If G has a Hamiltonian path, then for every S ⊂ V (G), c(G − S) ≤ |S| + 1. Lemma 4 ([19]). For a connected split graph G with ∆ I = 3, let v ∈ K, d I (v) = 3, and U = N I (v). If G is K 1,4-free, then N (U) = K. Corollary 1 (of Lemma 4). Let G be a connected K 1,4-free split graph with v ∈ K, d I (v) = 3. For every vertex w ∈ K \ {v}, N I (v) ∩ N I (w) = ∅. 2.1 Results on K 1,3-free split graphs Theorem 2. Let G be a connected K 1,3-free split graph. G contains a Hamiltonian path if and only if G has at most 2 vertices u, v ∈ I such that d(u) = 1, and d(v) = 1. Proof. If there exists at least three vertices {u, v, w} ⊆ I such that d(u) = d(v) = d(w) = 1, then clearly G has no Hamiltonian path. For the sufficiency, we see the following cases. Case 1: For every u ∈ I, if d(u) ≥ 2, then G is 2-connected, and by Lemma 2, G has a Hamiltonian cycle. Thus G has a Hamiltonian path. Case 2: If there exists only one vertex u ∈ I with d(u) = 1, then observe that G − u is 2-connected. By Lemma 2, there is a Hamiltonian cycle in G − u, which can be easily extended to a Hamiltonian path in G. Case 3: There exists two vertices u, v ∈ I with d(u) = d(v) = 1. If I = {u, v}, it is easy to see that there is a (u, v)-Hamiltonian path in G. If I = {u, v, w}, then we claim that N (w) ∩ N (u) = ∅ and N (w) ∩ N (v) = ∅. Suppose N (w) ∩ N (u) = ∅, then let N I (u ′) = {u, w}, u ′ ∈ K. Clearly, all the vertices x ∈ K \ {u ′ } are adjacent to w, otherwise {u ′ , u, w, x} induces a K 1,3. It follows that K ∪ {w} is a clique of larger size, contradicting the maximality of K. Similar arguments hold with respect to the vertex v, and hence N (w) ∩ N (v) = ∅. Thus we conclude that ∆ I = 1. From Lemma 1, if |I| > 3, since G is connected, ∆ I = 1. Now we produce a Hamiltonian path in G with |I| ≥ 3 as follows. Let I = {u, v, w 1 ,. .. , w k }, k ≥ 1 such that for all w i , 1 ≤ i ≤ k, d(w i) > 1. Let x i , y i , 1 ≤ i ≤ k be any two elements in N (w i). Since ∆ I = 1, note that for all s, t ∈ I, N (s) ∩ N (t) = ∅. Let P

2002

Many conditions implying the existence of a Hamiltonian cycle in a graph have been generalized to yield the existence of k edge-disjoint such cycles (see [3], [2], [5], [1] and [6]). In [7], Moon and Moser have shown that if G = (X,Y ) is a bipartite graph with |X| = |Y | = n such that for any non-adjacent pair of vertices (x, y) ∈ V (X)× V (Y ), d(x) + d(y) ≥ n + 1, then G has a Hamiltonian cycle. In this paper, we prove that if the degree condition is replaced by d(x) + d(y) ≥ n + 2k − 1, (for a given natural number k), then G contains k edgedisjoint Hamiltonian Cycles. 1 Definitions and notation Let G = (X, Y ) be a balanced bipartite graph. For two given subgraphs F and H of G, we denote by FH the subgraph of G induced by the vertices of F in H (i.e. the graph with vertex set V (F ) and edge set E(F ) ∩ E(H). For any given subgraphs A and B of G, let N(A) be the set of vertices of G that are adjacent to at least one vertex of A, let E(A,B) be the set of edges that have an end ve...

Commentarii Mathematici Helvetici, 1969

1979

The notions not defined here will be used in the sense of [1]. We shall show a recursive formula for the number of hamiltonian cycles of the graph K(m u m 2 , ..., m n ). In the case m x = m 2 = ... = m n = 2 we prove an explicit formula. Further we give an upper estimation for the number of hamiltonian cycles of an arbitrary graph G. A complete n -partite graph G is a graph whose vertex set V can be partitioned into n mutually disjoint subsets V u V 2 , ..., V n , whose union is V such that two vertices are connected by an edge if and only if they belong to different parts of the partition. If |Vi|=Wi, \V 2 \=m 2 , ..., \V n \=m n , we write G = K(m u m 2 , ..., m n ). The number of hamiltonian cycles of a graph G will be denote by H(G), in the case of the complete rz-partite graph K(m u m 2 , ...,m n )by H(m u m 2 ,..., m n , n).

arXiv: Combinatorics, 2020

Let $G$ be a simple graph of order $n$. The double vertex graph $F_2(G)$ of $G$ is the graph whose vertices are the $2$-subsets of $V(G)$, where two vertices are adjacent in $F_2(G)$ if their symmetric difference is a pair of adjacent vertices in $G$. A generalization of this graph is the complete double vertex graph $M_2(G)$ of $G$, defined as the graph whose vertices are the $2$-multisubsets of $V(G)$, and two of such vertices are adjacent in $M_2(G)$ if their symmetric difference (as multisets) is a pair of adjacent vertices in $G$. In this paper we exhibit an infinite family of graphs (containing Hamiltonian and non-Hamiltonian graphs) for which $F_2(G)$ and $M_2(G)$ are Hamiltonian. This family of graphs is the set of join graphs $G=G_1 + G_2$, where $G_1$ and $G_2$ are of order $m\geq 1$ and $n\geq 2$, respectively, and $G_2$ has a Hamiltonian path. For this family of graphs, we show that if $m\leq 2n$ then $F_2(G)$ is Hamiltonian, and if $m\leq 2(n-1)$ then $M_2(G)$ is Hamilt...

Fundamental Journal of Mathematics and Applications, 2018

A graph is called Hamiltonian (resp. traceable) if the graph has a Hamiltonian cycle (resp. path), a cycle (resp. path) containing all the vertices of the graph. In this note, we present sufficient conditions involving minimum degree and size for Hamiltonian and traceable graphs. One of the sufficient conditions strengthens the result obtained by Nikoghosyan in [1].

Zenodo (CERN European Organization for Nuclear Research), 2013

Let n ≥ 2 be an integer. The complete graph Kn with 1-factor I added has a decomposition into Hamilton cycles if and only if n is even. We show that Kn + I has a decomposition into Hamilton cycles which are symmetric with respect to the 1-factor I added. We also show that the complete bipartite graph Kn,n plus a 1-factor has a symmetric Hamilton decomposition, where n is odd.

Matemática Contemporânea, 2022

The complete double vertex graph M2(G) of G is defined as the graph whose vertices are the 2-multisubsets of V (G), and two of such vertices are adjacent in M2(G) if their symmetric difference (as multisets) is a pair of adjacent vertices in G. In this paper we exhibit an infinite family of graphs G (containing Hamiltonian and non-Hamiltonian graphs) for which M2(G) are Hamiltonian.