Subgraph conditions for Hamiltonian properties of graphs

The research that forms the basis of this thesis addresses the following general structural questions in graph theory: which fixed graph of pair of graphs do we have to forbid as an induced subgraph of an arbitrary graph G to guarantee that G has a nice structure?\ud In this thesis the nice structural property we have been aiming for is the existence of a Hamilton cycle, i.e., a cycle containing all the vertices of the graph, or related properties like the existence of a Hamilton path, of cycles of every length, or of Hamilton paths starting at every vertex of the graph. For these structural properties, sufficient Ore-type degree conditions are known since the 1960s. These Ore-conditions are of the type: if every pair of nonadjacent vertices of the graph G has degree sum at least some lower bound, the G is guaranteed to have the structural property. In order to obtain common generalizations of these sufficiency results based on Ore-type degree sum conditions on one hand and forbidden induced subgraph conditions on the other hand, the following questions have also been addressed in the thesis. Can we restrict the corresponding Ore-type degree sum condition to certain induced subgraphs of pairs of induced subgraphs of a graph G and still guarantee that G has the same nice structure? In the thesis work we have proved many examples that provide affirmative answers to these general questions. We refer to the listed chapters for the details and the the precise definitions and formulations of the results

On path-quasar Ramsey numbers

Let $G_1$ and $G_2$ be two given graphs. The Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\overline{G}$ contains a $G_2$. Parsons gave a recursive formula to determine the values of $R(P_n,K_{1,m})$, where $P_n$ is a path on $n$ vertices and $K_{1,m}$ is a star on $m+1$ vertices. In this note, we first give an explicit formula for the path-star Ramsey numbers. Secondly, we study the Ramsey numbers $R(P_n,K_1\vee F_m)$, where $F_m$ is a linear forest on $m$ vertices. We determine the exact values of $R(P_n,K_1\vee F_m)$ for the cases $m\leq n$ and $m\geq 2n$, and for the case that $F_m$ has no odd component. Moreover, we give a lower bound and an upper bound for the case $n+1\leq m\leq 2n-1$ and $F_m$ has at least one odd component.Comment: 7 page

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