Heavy subgraphs, stability and hamiltonicity

Let $G$ be a graph. Adopting the terminology of Broersma et al. and \v{C}ada, respectively, we say that $G$ is 2-heavy if every induced claw ($K_{1,3}$) of $G$ contains two end-vertices each one has degree at least $|V(G)|/2$; and $G$ is o-heavy if every induced claw of $G$ contains two end-vertices with degree sum at least $|V(G)|$ in $G$. In this paper, we introduce a new concept, and say that $G$ is \emph{$S$-c-heavy} if for a given graph $S$ and every induced subgraph $G'$ of $G$ isomorphic to $S$ and every maximal clique $C$ of $G'$, every non-trivial component of $G'-C$ contains a vertex of degree at least $|V(G)|/2$ in $G$. In terms of this concept, our original motivation that a theorem of Hu in 1999 can be stated as every 2-connected 2-heavy and $N$-c-heavy graph is hamiltonian, where $N$ is the graph obtained from a triangle by adding three disjoint pendant edges. In this paper, we will characterize all connected graphs $S$ such that every 2-connected o-heavy and $S$-c-heavy graph is hamiltonian. Our work results in a different proof of a stronger version of Hu's theorem. Furthermore, our main result improves or extends several previous results.Comment: 21 pages, 6 figures, finial version for publication in Discussiones Mathematicae Graph Theor

Neighborhood intersections and Hamiltonicity in almost claw-free graphs

AbstractLet G be a graph. The partially square graph G∗ of G is a graph obtained from G by adding edges uv satisfying the conditions uv∉E(G), and there is some w∈N(u)∩N(v), such that N(w)⊆N(u)∪N(v)∪{u,v}. Let t>1 be an integer and Y⊆V(G), denote n(Y)=|{v∈V(G)|miny∈Y{distG(v,y)}⩽2}|,It(G)={Z|Z is an independent set of G,|Z|=t}. In this paper, we show that a k-connected almost claw-free graph with k⩾2 is hamiltonian if ∑z∈Zd(z)⩾n(Z)−k in G for each Z∈Ik+1(G∗), thereby solving a conjecture proposed by Broersma, Ryjác̆ek and Schiermeyer. Zhang's result is also generalized by the new result

Ore- and Fan-type heavy subgraphs for Hamiltonicity of 2-connected graphs

Bedrossian characterized all pairs of forbidden subgraphs for a 2-connected graph to be Hamiltonian. Instead of forbidding some induced subgraphs, we relax the conditions for graphs to be Hamiltonian by restricting Ore- and Fan-type degree conditions on these induced subgraphs. Let $G$ be a graph on $n$ vertices and $H$ be an induced subgraph of $G$. $H$ is called \emph{o}-heavy if there are two nonadjacent vertices in $H$ with degree sum at least $n$, and is called $f$-heavy if for every two vertices $u,v\in V(H)$, $d_{H}(u,v)=2$ implies that $\max\{d(u),d(v)\}\geq n/2$. We say that $G$ is $H$-\emph{o}-heavy ($H$-\emph{f}-heavy) if every induced subgraph of $G$ isomorphic to $H$ is \emph{o}-heavy (\emph{f}-heavy). In this paper we characterize all connected graphs $R$ and $S$ other than $P_3$ such that every 2-connected $R$-\emph{f}-heavy and $S$-\emph{f}-heavy ($R$-\emph{o}-heavy and $S$-\emph{f}-heavy, $R$-\emph{f}-heavy and $S$-free) graph is Hamiltonian. Our results extend several previous theorems on forbidden subgraph conditions and heavy subgraph conditions for Hamiltonicity of 2-connected graphs.Comment: 21 pages, 2 figure

On hamiltonicity of 1-tough triangle-free graphs

Let ω(G) denote the number of components of a graph G. A connected graph G is said to be 1-tough if ω(G − X)≤|X| for all X ⊆ V(G) with ω(G − X)&gt;1. It is well-known that every hamiltonian graph is 1-tough, but that the reverse statement is not true in general, and even not for triangle-free graphs. We present two classes of triangle-free graphs for which the reverse statement holds, i.e., for which hamiltonicity and 1-toughness are equivalent. Our two main results give partial answers to two conjectures due to Nikoghosyan.</p

Connected Domination Critical Graphs

This thesis investigates the structure of connected domination critical graphs. The characterizations developed provide an important theoretical framework for addressing a number of difficult computational problems in the areas of operations research (for example, facility locations, industrial production systems), security, communication and wireless networks, transportation and logistics networks, land surveying and computational biology. In these application areas, the problems of interest are modelled by networks and graph parameters such as domination numbers reflect the efficiency and performance of the systems