A note on maximal nonhamiltonian Burkard-Hammer graphs

Discussiones Mathematicae Graph Theory, 2004

A graph G is called a split graph if the vertex-set V of G can be partitioned into two subsets V 1 and V 2 such that the subgraphs of G induced by V 1 and V 2 are empty and complete, respectively. In this paper, we characterize hamiltonian graphs in the class of split graphs with minimum degree δ at least |V 1 | − 2.

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...

2019

For a set of graphs F , let H(n;F) denote the class of non-bipartite Hamiltonian graphs on n vertices that does not contain any graph of F as a subgraph and h(n;F) = max{E(G) : G ∈ H(n;F)} where E(G) is the number of edges in G. In this paper, we determine h(n; {θ4, θ5, θ7}) and we establish an upper bound of h(n; θ7) for sufficiently even large n. Our results confirms the conjecture made in [1] for k = 3. ∗. Corresponding author 414 M.S. BATAINEH, A.A. AL-RHAYYEL, ZEAD MUSTAFA and M.M.M. JARADAT

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.

Discrete Mathematics, 1996

Related to Chvfital's famous conjecture stating that every 2-tough graph is hamiltonian, we study the relation of toughness and hamiltonieity on special classes of graphs. First, we consider properties of graph classes related to hamiltonicity, traceability and toughness concepts and display some algorithmic consequences. Furthermore, we present a polynomial time algorithm deciding whether the toughness of a given split graph is less than one and show that deciding whether the toughness of a bipartite graph is exactly one is coNP-complete. We show that every 3-tough split graph is hamiltonian and that there is a sequence of nonhamiltonian split graphs with toughness converging to 3.

For integer k ≥ 1, a (p, q)-graph G = (V, E) is said to admit an AL(k)-traversal if there exist a sequence of vertices (v1, v2, . . . , vp) such that for each i = 1, 2, . . . , p − 1, the distance between vi and vi+1 is k. We call a graph k-step Hamiltonian (or admits a k-step Hamiltonian tour) if it admits an AL(k)-traversal and d(v1, vp) = k. In this paper we consider k-step Hamiltonicity of bipartite and tripartite graphs. As an application, we found that a 2-step Hamiltonian tour of a graph could sometimes induce a super-edge-magic labeling of the graph.

Mathematical Problems of Computer Science, 2018

R. Wang (Discrete Mathematics and Theoretical Computer Science, vol. 19(3), 2017) proposed the following problem. Problem. Let D be a strongly connected balanced bipartite directed graph of order 2a ≥8. Suppose that d(x) ≥ 2a - k, d(y) ≥a + k or d(y) ≥2a - k, d(x) ≥a + k for every pair of vertices {x; y}with a common out-neighbour, where 2 • k • a=2. Is D Hamiltonian? In this paper, we prove that if a digraph D satis¯es the conditions of this problem, then (i) D contains a cycle factor, (ii) for every vertex x ∈ V (D) there exists a vertex y ∈ V (D) such that x and y have a common out-neighbour.

Journal of the Indonesian Mathematical Society

A split graph is a graph in which the vertices can be partitioned into an independent set and a clique. A graph is split if and only if it has no induced subgraph isomorphic to C5, C4 or 2K2, which is a well-known characterization for split graph. A property of a graph G is recognizable if it can be recognized from the collection of all maximal proper induced subgraphs of G. We show that any nonsplit graph can have at most five split maximal induced subgraphs. Also we list out all C5-free nonsplit graphs having split maximal induced subgraphs, which is the main and, in fact, tedious result of this paper

Information Processing Letters, 2006

Rahman and Kaykobad proved the following theorem on Hamiltonian paths in graphs. Let G be a connected graph with n vertices. If d(u) + d(v) + δ(u, v) n + 1 for each pair of distinct non-adjacent vertices u and v in G, where δ(u, v) is the length of a shortest path between u and v in G, then G has a Hamiltonian path. It is shown that except for two families of graphs a graph is Hamiltonian if it satisfies the condition in Rahman and Kaykobad's theorem. The result obtained in this note is also an answer for a question posed by Rahman and Kaykobad.

Discrete Mathematics, 2002

Let G = (X; Y) be a 2-connected balanced bipartite graph with |X | = |Y | = n. In this paper, we prove that if |N (x1) ∪ N (x2)| + |N (y1) ∪ N (y2)| ¿ n + 2 for any {x1; x2} ⊆ X and {y1; y2} ⊆ Y , then G is hamiltonian except when G is a special graph on 8 or on 12 vertices.