Related papers: Graphs with many strong orientations

A graph $G$ is $k$-path-coverable if its vertex set $V(G)$ can be covered by $k$ or fewer vertex disjoint paths. In this paper, using the $Q$-index of a connected graph $G$, we present a tight sufficient condition for $G$ with fixed minimum…

Let $d\geq 3$ be a constant and let $F$ be a $d$-regular graph on $[n]$ with not too many symmetries. By the union bound, the probability threshold for the existence of a spanning subgraph in $G(n,p)$ isomorphic to $F$ is at least…

For many types of graphs, criteria have been discovered that give necessary and sufficient conditions for an integer sequence to be the degree sequence of such a graph. These criteria tend to take the form of a set of inequalities, and in…

Let $G$ be a connected graph on $n$ vertices and at most $n(1+\epsilon)$ edges with bounded maximum degree, and $F$ a graph on $n$ vertices with minimum degree at least $n-k$, where $\epsilon$ is a constant depending on $k$. In this paper,…

Let s<t be two fixed positive integers. We study what are the minimum degree conditions for a bipartite graph G, with both color classes of size n=k(s+t), which ensure that G has a K_{s,t}-factor. Exact result for large n is given. Our…

Let $G=(V(G),E(G))$ be a simple graph, where $V(G)$ and $E(G)$ are the vertex set and the edge set of $G$, respectively. The number of components of $G$ is denoted by $c(G)$. Let $t$ be a positive real number, and a connected graph $G$ is…

A graph $G=(V,E)$ is called $d$-rigid if, for a generic embedding of its vertices in $\mathbb{R}^d$, every edge-length preserving continuous motion of the vertices preserves the distances between all pairs of non-adjacent vertices as well.…

An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its neighbourhood within the identifying code. If $\M(G)$ denotes the minimum size of an identifying code of a graph $G$, it was…

An obstacle representation of a graph $G$ is a set of points in the plane representing the vertices of $G$, together with a set of polygonal obstacles such that two vertices of $G$ are connected by an edge in $G$ if and only if the line…

As a non-trivial extension of the celebrated Cheeger inequality, the higher-order Cheeger inequalities for graphs due to Lee, Oveis Gharan and Trevisan provide for each $k$ an upper bound for the $k$-way Cheeger constant in forms of…

Given a graph $G$, the strong clique number of $G$, denoted $\omega_S(G)$, is the maximum size of a set $S$ of edges such that every pair of edges in $S$ has distance at most $2$ in the line graph of $G$. As a relaxation of the renowned…

Consider an acyclic directed network $G$ with sources $S_1, S_2,..., S_l$ and distinct sinks $R_1, R_2,..., R_l$. For $i=1, 2,..., l$, let $c_i$ denote the min-cut between $S_i$ and $R_i$. Then, by Menger's theorem, there exists a group of…

We introduce a set of multi-way dual Cheeger constants and prove universal higher-order dual Cheeger inequalities for eigenvalues of normalized Laplace operators on weighted finite graphs. Our proof proposes a new spectral clustering…

An odd coloring of a graph $G$ is a proper coloring of $G$ such that for every non-isolated vertex $v$, there is a color appearing an odd number of times in $N_G(v)$. Odd coloring of graphs was studied intensively in recent few years. In…

We establish tight lower and upper bounds on the number of edges in traceable graphs in several classes of dense graphs. A graph is traceable if it has a Hamiltonian path. We show that the bound is: - quadratic for the class of graphs of…

A mixed multigraph is a multigraph which may contain both undirected and directed edges. An orientation of a mixed multigraph $G$ is an assignment of exactly one direction to each undirected edge of $G$. A mixed multigraph $G$ can be…

Consider a graph $G$ with a path $P$ of order $n$. What conditions force $G$ to also have a long induced path? As complete bipartite graphs have long paths but no long induced paths, a natural restriction is to forbid some fixed complete…

We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1-\epsilon)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d\geq 2…

We prove that there exists an absolute constant $C>0$ such that, for any positive integer $k$, every graph $G$ with minimum degree at least $Ck$ admits a vertex-partition $V(G)=S\cup T$, where both $G[S]$ and $G[T]$ have minimum degree at…

The conflict-free chromatic index of a graph $G$ is the minimum number of colours in an edge colouring of $G$ such that the neighbourhood of every edge contains a colour appearing exactly once. Its vertex analogue is the conflict-free…