Related papers: Absolutely avoidable order-size pairs for induced …

Let $G=(V,E)$ be an undirected graph with $n$ vertices and $m$ edges, in which each vertex $u$ is assigned an integer priority in $[1,n]$, with 1 being the "highest" priority. Let $M$ be a matching of $G$. We define the priority score of…

In this note we present an infinite family of 2-connected graphs such that their matching polynomials have only integer zeros. This answers in negative a question of Akbari et al. [Graphs with integer matching polynomial zeros, Discrete…

By a sphere-packing argument, we show that there are infinitely many pairs of primes that are close to each other for some metrics on the integers. In particular, for any numeration basis $q$, we show that there are infinitely many pairs of…

Given a family $\mathcal{F}$ of subsets of $[n]$, we say two sets $A, B \in \mathcal{F}$ are comparable if $A \subset B$ or $B \subset A$. Sperner's celebrated theorem gives the size of the largest family without any comparable pairs. This…

An edge of a quasi $k$-connected graph is said to be quasi $k$-contractible if the contraction of the edge results in a quasi $k$-connected graph. If every quasi $k$-connected graph without a quasi $k$-contractible edge has either $H_{1}$…

Let $f(m)$ be the largest integer such that for every set $A = \{a_1 < \cdots < a_m\}$ of $m$ positive integers and every open interval $I$ of length $2a_m$, there exist at least $f(m)$ disjoint pairs $(a, b)$ with $a \in A$ dividing $b \in…

Given a graph $G=(V,E)$, a subset $X$ of $V$ is an interval of $G$ provided that for any $a, b\in X$ and $ x\in V \setminus X$, $\{a,x\}\in E$ if and only if $\{b,x\}\in E$. For example, $\emptyset$, $\{x\}(x\in V)$ and $V$ are intervals of…

Let $M$ be a $3$-connected matroid. A pair $\{e,f\}$ in $M$ is detachable if $M \backslash e \backslash f$ or $M / e / f$ is $3$-connected. Williams (2015) proved that if $M$ has at least 13 elements, then at least one of the following…

Let $G=(V,E)$ be an undirected graph without loops and multiple edges. A subset $C\subseteq V$ is called \emph{identifying} if for every vertex $x\in V$ the intersection of $C$ and the closed neighbourhood of $x$ is nonempty, and these…

Given a graph $G$ on $n$ vertices, for which $m$ is it possible to partition the edge set of the $m$-fold complete graph $mK_n$ into copies of $G$? We show that there is an integer $m_0$, which we call the \emph{partition modulus of $G$},…

We prove a recent conjecture of Beisegel et al. that for every positive integer k, every graph containing an induced P_k also contains an avoidable P_k. Avoidability generalises the notion of simpliciality best known in the context of…

We construct a symmetric invertible binary pairing function $F(m,n)$ on the set of positive integers with a property of $F(m,n)=F(n,m)$. Then we provide a complete proof of its symmetry and bijectivity, from which the construction of…

In the course of proving the strong perfect graph theorem, Chudnovsky, Robertson, Seymour, and Thomas showed that every perfect graph either belongs to one of five basic classes or admits one of several decompositions. Four of the basic…

A tight Heffter array H(m,n) is an m x n matrix with nonzero entries from Z_{2mn+1} such that i) the sum of the elements in each row and each column is 0, and ii) no element from {x,-x\ appears twice. We prove that H(m,n) exist if and only…

A vertex labeling of a hypergraph is sum distinguishing if it uses positive integers and the sums of labels taken over the distinct hyperedges are distinct. Let s(H) be the smallest integer N such that there is a sum-distinguishing labeling…

For a graph $G$ with order $2n$ and a perfect matching, let $f(G)$ and $F(G)$ denote the minimum and maximum forcing number of $G$ respectively. Then $0\leq f(G)\leq F(G)\leq n-1$. Liu and Zhang [10] ever proposed a conjecture: $e(G)\geq…

A graph on $2k$ vertices is path-pairable if for any pairing of the vertices the pairs can be joined by edge-disjoint paths. The so far known families of path-pairable graphs have diameter of length at most 3. In this paper we present an…

The well-known Erd\H{o}s-Hajnal conjecture states that for any graph $F$, there exists $\epsilon>0$ such that every $n$-vertex graph $G$ that contains no induced copy of $F$ has a homogeneous set of size at least $n^{\epsilon}$. We consider…

In this paper, we develop efficient exact and approximate algorithms for computing a maximum independent set in random graphs. In a random graph $G$, each pair of vertices are joined by an edge with a probability $p$, where $p$ is a…

In 1930, Ramsey proved that every large graph contains either a large clique or a large edgeless graph as an induced subgraph. It is well known that every large connected graph contains a long path, a large clique, or a large star as an…