Related papers: Spanning path-cycle systems with given end-vertice…

Let $G$ be a $k$-connected graph with $k\geq 2$. In this paper we first prove that: For two distinct vertices $x$ and $z$ in $G$, it contains a path passing through its any $k-2$ {specified} vertices with length at least the average degree…

In this paper, we show that for any positive integer $m$ and $k\in [2]$, let $G$ be a $(2m+2k+2)$-connected graph and let $a_1,\ldots , a_m, s, t$ be any distinct vertices of $G$, there are $k$ internally disjoint $s$-$t$ paths $P_1,…

Given an undirected graph $G=(V,E)$ and vertices $s,t,w_1,w_2\in V$, we study finding whether there exists a simple path $P$ from $s$ to $t$ such that $w_1,w_2 \in P$. As a sub-problem, we study the question: given an undirected graph and…

For a simple graph $G$, let $n$ and $m$ denote the number of vertices and edges in $G$, respectively. The Erd\H{o}s-Gallai theorem for paths states that in a simple $P_k$-free graph, $m \leq \frac{n(k-1)}{2}$, where $P_k$ denotes a path…

Let $c\in (0, 1]$ be a real number and let $n$ be a sufficiently large integer. We prove that every $n$-vertex $c n$-regular graph $G$ contains a collection of $\lfloor 1/c \rfloor$ paths whose union covers all but at most $o(n)$ vertices…

The cycles are the only $2$-connected graphs in which any two nonadjacent vertices form a vertex cut. We generalize this fact by proving that for every integer $k\ge 3$ there exists a unique graph $G$ satisfying the following conditions:…

Let $G$ be a graph, and $v\in V(G)$ and $S\subseteq V(G)\backslash v$ of size at least $k$. An important result on graph connectivity due to Perfect states that, if $v$ and $S$ are $k$-linked, then a $(k-1)$-link between a vertex $v$ and…

We generalise a result of Corr\'{a}di and Hajnal and show that every graph with average degree at least $\tfrac{4}{3}kr$ contains $k$ vertex disjoint cycles, each of order at least $r$, as long as $k \geq 6$. This bound is sharp when $r=3$.

In this paper we provide some sufficient conditions for the existence of an odd or even cycle that passing a given vertex or an edge in $2$-connected or $2$-edge connected graphs. We provide some similar conditions for the existence of an…

A $3$-connected graph $G$ is essentially $4$-connected if, for any $3$-cut $S\subseteq V(G)$ of $G$, at most one component of $G-S$ contains at least two vertices. We prove that every essentially $4$-connected maximal planar graph $G$ on…

For a $2$-connected graph $G$ and vertices $u,v$ of $G$ we define an abstract graph $\mathcal{P}(G_{uv})$ whose vertices are the paths joining $u$ and $v$ in $G$, where paths $S$ and $T$ are adjacent if $T$ is obtained from $S$ by replacing…

Menger's Edge Theorem asserts that there exist $k$ pairwise edge-disjoint paths between two vertices in an undirected graph if and only if a deletion of any $k-1$ or less edges does not disconnect these two vertices. Alternatively, there…

Let $k \ge 3$ be an integer, $H_{k}(G)$ be the set of vertices of degree at least $2k$ in a graph $G$, and $L_{k}(G)$ be the set of vertices of degree at most $2k-2$ in $G$. In 1963, Dirac and Erd\H{o}s proved that $G$ contains $k$…

Let $H$ be obtained from a cyclically $4$-edge-connected cubic planar graph $Y$ other than $K_4$ by deleting two adjacent vertices. We provide a short proof that if $H$ has circumference at least $k$ for some even integer $k \ge 4$, then…

Let us say a graph is $s\mathcal{O}$-free, where $s\ge 1$ is an integer, if there do not exist $s$ cycles of the graph that are pairwise vertex-disjoint and have no edges joining them. The structure of such graphs, even when $s=2$, is not…

Let $\mathscr{G}$ be the class of plane graphs without triangles normally adjacent to $8^{-}$-cycles, without $4$-cycles normally adjacent to $6^{-}$-cycles, and without normally adjacent $5$-cycles. In this paper, it is shown that every…

Bondy and Vince showed that every graph with minimum degree at least three contains two cycles of lengths differing by one or two.We prove the following average degree counterpart that every $n$-vertex graph $G$ with at least $\frac52(n-1)$…

Lov\'asz (1965) characterized graphs without two vertex-disjoint cycles, which implies that such graphs have at most three vertices hitting all cycles. In this paper, we ask whether such a small hitting set exists for $S$-cycles, when a…

We conjecture that the balanced complete bipartite graph $K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil}$ contains more cycles than any other $n$-vertex triangle-free graph, and we make some progress toward proving this. We give equivalent…

Let $k \geq 3$ be an integer, $h_{k}(G)$ be the number of vertices of degree at least $2k$ in a graph $G$, and $\ell_{k}(G)$ be the number of vertices of degree at most $2k-2$ in $G$. Dirac and Erd\H{o}s proved in 1963 that if $h_{k}(G) -…