相关论文: Star-factors of tournaments

In this paper, we study $(1,2)$-step competition graphs of bipartite tournaments. A bipartite tournament means an orientation of a complete bipartite graph. We show that the $(1,2)$-step competition graph of a bipartite tournament has at…

We prove that the order of the double reduced 2-factor transfer digraph ${\cal R}^{**}_{m}$ which is needed for the enumeration of the spanning unions of cycles in the rectangular grid graph $P_m \times P_n$ ($m,n \in N$), when $m$ is odd,…

Let $\mathrm{rex}(n, F)$ denote the maximum number of edges in an $n$-vertex graph that is regular and does not contain $F$ as a subgraph. We give lower bounds on $\mathrm{rex}(n, F)$, that are best possible up to a constant factor, when…

We study a game on a graph $G$ played by $r$ {\it revolutionaries} and $s$ {\it spies}. Initially, revolutionaries and then spies occupy vertices. In each subsequent round, each revolutionary may move to a neighboring vertex or not move,…

We show that for every $n\in\mathbb N$ and $\log n\le d\le n$, if a graph $G$ has $N=\Theta(dn)$ vertices and minimum degree $(1+o(1))\frac{N}{2}$, then it contains a spanning subdivision of every $n$-vertex $d$-regular graph.

A tournament is an orientation of a complete graph. We say that a vertex $x$ in a tournament $\vec T$ controls another vertex $y$ if there exists a directed path of length at most two from $x$ to $y$. A vertex is called a king if it…

Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is $m$-\textit{coloured} if each of the $m$ colours is used. For an $m$-colouring $\Delta$ of $\mathbb{N}^{(2)}$, the complete graph on $\mathbb{N}$, we…

A line digraph $L(G) = (A, E)$ is the digraph constructed from the digraph $G = (V, A)$ such that there is an arc $(a,b)$ in $L(G)$ if the terminal node of $a$ in $G$ is the initial node of $b$. The maximum number of arcs in a line digraph…

In this paper we explore some results concerning the spread of the line and the total graph of a given graph. In particular, it is proved that for an $(n,m)$ connected graph $G$ with $m > n \geq 4$ the spread of $G$ is less than or equal to…

A sequence $S$ is potentially $K_{m}-C_{4}$-graphical if it has a realization containing a $K_{m}-C_{4}$ as a subgraph. Let $\sigma(K_{m}-C_{4}, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with…

We prove Seymour's Second Neighborhood Conjecture when the missing graph is disjoint stars under some conditions. Weaker conditions are required when n=2 or 3. In some cases, we exhibit two vertices with the desired property.

Let m(G) be the maximum number of vertex-disjoint odd cycles of a graph G and t(G) the minimum number of vertices whose removal makes G bipartite. We show that t(G)<=6m(G) if G is planar. This improves the previous bound t(G)<=10m(G) by…

Let $G$ be a graph. We denote by $e(G)$ and $\rho(G)$ the size and the spectral radius of $G$. A spanning subgraph $F$ of $G$ is called an even factor of $G$ if $d_F(v)\in\{2,4,6,\ldots\}$ for every $v\in V(G)$. Yan and Kano provided a…

Sullivan stated the conjectures: (1) every oriented graph $D$ has a vertex $x$ such that $d^{++}(x)\geq d^{-}(x)$; (2) every oriented graph $D$ has a vertex $x$ such that $d^{++}(x)+d^{+}(x)\geq 2d^{-}(x)$. In this paper, we prove that…

In this paper, we prove that all but one of the components of the transfer digraph ${\cal D}^*_m$ needed for the enumeration of 2-factors in the rectangular, thick cylinder and Moebius strip grid graphs of the fixed width $m$ $(m \in N)$…

Let $\Ga$ be the collection of all weighted bipartite graphs each having $\sigma$ and $m$, as the size of a vertex partition and the total weight, respectively. We give a tight lower bound $\lceil \frac{m-\sigma}{\sigma} \rceil+1$ for the…

We prove that every $n$-vertex directed graph $G$ with the minimum outdegree $\delta^+(G) = d$ contains a subgraph $H$ satisfying \[ \min\left\{\delta^+(H), \delta^-(H) \right\} \ge \frac{d(d+1)}{2n} \,.\] We also show that if $d = o(n)$…

Given a positive integer $m\ge 3$, let $ch(m)$ be the smallest positive constant with the following property: \emph{ Every simple directed graph on $n\ge 3$ vertices all whose outdegrees are at least $ch(m)\cdot n$ contains a directed cycle…

The convex hull of the set of the incidence vectors of the matchings of a graph G is the matching polytope of the graph, M(G). The graph whose vertices and edges are the vertices and edges of M(G) is the skeleton of the matching polytope of…

Let k_r(n,m) denote the minimum number of r-cliques in graphs with n vertices and m edges. For r=3,4 we give a lower bound on k_r(n,m) that approximates k_r(n,m) with an error smaller than n^r/(n^2-2m). The solution is based on a constraint…