116 research outputs found

### Long dominating cycles and paths in graphs with large neighborhood unions

Let G be a graph of order n and define NC(G) = min{|N(u)U N(v)| |uv E(G)}. A cycle C of G is called a dominating cycle or D-cycle if V(G) - V(C) is an independent set. A D-path is defined analogously. The following result is proved: if G is 2-connected and contains a D-cycle, then G contains a D-cycle of length at least min{n, 2NC(G)} unless G is the Petersen graph. By combining this result with a known sufficient condition for the existence of a D-cycle, a common generalization of Ore's Theorem and several recent neighborhood union results is obtained. An analogous result on long D-paths is also established

### Path-kipas Ramsey numbers

For two given graphs $F$ and $H$, the Ramsey number $R(F,H)$ is the smallest positive integer $p$ such that for every graph $G$ on $p$ vertices the following holds: either $G$ contains $F$ as a subgraph or the complement of $G$ contains $H$ as a subgraph. In this paper, we study the Ramsey numbers $R(P_n,\hat{K}_m)$, where $P_n$ is a path on $n$ vertices and $\hat{K}_m$ is the graph obtained from the join of $K_1$ and $P_m$. We determine the exact values of $R(P_n,\hat{K}_m)$ for the following values of $n$ and $m$: $1\le n \le 5$ and $m\ge 3$; $n\ge 6$ and ($m$ is odd, $3\le m\le 2n-1$) or ($m$ is even, $4\le m \le n+1$); $6\le n\le 7$ and $m=2n-2$ or $m \ge 2n$; $n\ge 8$ and $m=2n-2$ or $m=2n$ or ($q\cdot n-2q+1 \le m\le q\cdot n-q+2$ with $3\le q\le n-5$) or $m\ge (n-3)^2$; odd $n\ge 9$ and ($q\cdot n-3q+1\le m\le q\cdot n-2q$ with $3\le q\le (n-3)/2$) or ($q\cdot n-q-n+4m\le q\cdot n-2q$ with $(n-1)/2\le q\le n-4).$ Moreover, we give lower bounds and upper bounds for $R(P_n ,\hat{K}_m)$ for the other values of $m$ and $n$

### The Hamiltonian index of a graph and its branch-bonds

Let $G$ be an undirected and loopless finite graph that is not a path. The minimum $m$ such that the iterated line graph $L^m(G)$ is hamiltonian is called the hamiltonian index of $G,$ denoted by $h(G).$ A reduction method to determine the hamiltonian index of a graph $G$ with $h(G)\geq 2$ is given here. With it we will establish a sharp lower bound and a sharp upper bound for $h(G)$, respectively, which improves some known results of P.A. Catlin et al. [J. Graph Theory 14 (1990)] and H.-J. Lai [Discrete Mathematics 69 (1988)]. Examples show that $h(G)$ may reach all integers between the lower bound and the upper bound. \u

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