Related papers: The existence of a path-factor without small odd p…

A graph $G$ has the $k$-strong parity property if for any $X\subseteq V(G)$ with $|X|$ even, $G$ contains a spanning subgraph $F$ with $d_F(u)\equiv1$ (mod 2) for each $u\in X$ and $d_F(v)\in\{k,k+2,k+4,\ldots\}$ for each $v\in…

We study the least doubling constant $C_G$ among all possible doubling measures defined on a path graph $G$. We consider both finite and infinite cases and show that, if $G=\mathbb Z$, $C_{\mathbb Z}=3$, while for $G=L_n$, the path graph…

In this paper, we show that every $(3k-3)$-edge-connected graph $G$, under a certain condition on whose degrees, can be edge-decomposed into $k$ factors $G_1,\ldots, G_k$ such that for each vertex $v\in V(G_i)$, $|d_{G_i}(v)-d_G(v)/k|< 1$,…

For a subgraph $G$ of the blow-up of a graph $F$, we let $\delta^*(G)$ be the smallest minimum degree over all of the bipartite subgraphs of $G$ induced by pairs of parts that correspond to edges of $F$. In [Triangle-factors in a balanced…

In this paper we consider the cop number of graphs with no, or few, short cycles. We show that when $G$ is graph of girth $g$ and the minimum degree $\delta \geq 2$, then $c(G) = O(n\log(n)(\delta-1)^{-\lfloor \frac{g+1}{4} \rfloor})$ as a…

We call a set $\mathcal S$ of graphs an "even subdivison-factor" of a cubic graph $G$ if $G$ contains a spanning subgraph $H$ such that every component of $H$ has an even number of vertices and is a subdivision of an element of $\mathcal…

A graph $G$ is said to be equitably $c$-colorable if its vertices can be partitioned into $c$ independent sets that pairwise differ in size by at most one. Chen, Lih, and Wu conjectured that every connected graph $G$ with maximum degree…

A 2-factor of a graph $G$ is a 2-regular spanning subgraph of $G$. We present a survey summarising results on the structure of 2-factors in regular graphs, as achieved by various researchers in recent years.

We show that if G is a connected bridgeless cubic graph whose every 2-factor is comprised of cycles of length five then G is the Petersen graph.

Let $c$ denote the largest constant such that every $C_{6}$-free graph $G$ contains a bipartite and $C_4$-free subgraph having $c$ fraction of edges of $G$. Gy\H{o}ri et al. showed that $\frac{3}{8} \le c \le \frac{2}{5}$. We prove that…

A pseudo 2-factor of a graph is a spanning subgraph such that each component is $K_1$, $K_2$, or a cycle. This notion was introduced by Bekkai and Kouider in 2009, where they showed that every graph $G$ has a pseudo 2-factor with at most…

We prove that there exist functions $f$ and $g$ such that for all positive integers $k$ and $d$, for every graph $G$ and every subset $A$ of the vertices of $G$, either $G$ contains $k$ $A$-paths such that vertices of different $A$-paths…

Let $G$ be a graph. We say that $G$ is perfectly divisible if for each induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B])<\omega(H)$. We use $P_t$ and $C_t$ to denote a path…

We prove that for any graph G at least one of G or $\bar{G}$ satisfies $\chi \leq {1/4}\omega + {3/4}\Delta + 1$. In particular, self-complementary graphs satisfy this bound.

Any finite simple graph $G = (V,E)$ can be represented by a collection $\mathscr{C}$ of subsets of $V$ such that $uv\in E$ if and only if $u$ and $v$ appear together in an odd number of sets in $\mathscr{C}$. Let $c_2(G)$ denote the minimum…

Win [\emph{J. Graph Theory} {\bf 6}(1982), 489--492] conjectured that a graph $G$ on $n$ vertices contains $k$ disjoint perfect matchings, if the degree sum of any two nonadjacent vertices is at least $n+k-2$, where $n$ is even and $n\geq…

We say that a graph $G$ is $(2,m)$-linked if, for any distinct vertices $a_1,\ldots, a_m, b_1,b_2$ in $G$, there exist vertex disjoint connected subgraphs $A,B$ of $G$ such that $\{a_1, \ldots, a_m\}$ is contained in $A$ and $\{b_1,b_2\}$…

The distance signless Laplacian matrix of a graph $G$ is define as $Q(G)=$Tr$(G)+D(G)$, where Tr$(G)$ and $D(G)$ are the diagonal matrix of vertex transmissions and the distance matrix of $G$, respectively. Denote by $E_G(v)$ the set of all…

We first prove that for every vertex x of a 4-connected graph G there exists a subgraph H in G isomorphic to a subdivision of the complete graph K4 on four vertices such that G-V(H) is connected and contains x. This implies an affirmative…

For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of size at least $2$, a path in $G$ is said to be an $S$-path if it connects all vertices of $S$. Two $S$-paths $P_1$ and $P_2$ are said to be internally disjoint if $E(P_1)\cap…