Related papers: There are no excess one digraphs

In this paper, we investigate bounds for the following judicious $k$-partitioning problem: Given an edge-weighted graph $G$, find a $k$-partition $(V_1,V_2,\dots ,V_k)$ of $V(G)$ such that the total weight of edges in the heaviest induced…

Several domination results have been obtained for maximal outerplanar graphs (mops). The classical domination problem is to minimize the size of a set $S$ of vertices of an $n$-vertex graph $G$ such that $G - N[S]$, the graph obtained by…

The hypergraph Moore bound is an elegant statement that characterizes the extremal trade-off between the girth - the number of hyperedges in the smallest cycle or even cover (a subhypergraph with all degrees even) and size - the number of…

We consider drawings of graphs in the plane in which edges are represented by polygonal paths with at most one bend and the number of different slopes used by all segments of these paths is small. We prove that…

Mubayi and Verstraete conjectured that if $T$ is a tree on $t + 1$ vertices, then any $n$-vertex graph $G$ with average degree $d$ contains at least \[ n d(d - 1) \cdots (d - t + 1) \] labeled copies of $T$ as long as $d$ is sufficiently…

A graph $G = (V,E)$ is globally rigid in $\mathbb{R}^d$ if for any generic placement $p : V \rightarrow \mathbb{R}^d$ of the vertices, the edge lengths $||p(u) - p(v)||, uv \in E$ uniquely determine $p$, up to congruence. In this paper we…

We show that every directed graph with minimum out-degree at least $18k$ contains at least $k$ vertex disjoint cycles. This is an improvement over the result of Alon who showed this result for digraphs of minimum out-degree at least $64k$.…

A graph is $k$-planar if it can be drawn in the plane such that no edge is crossed more than $k$ times. While for $k=1$, optimal $1$-planar graphs, i.e., those with $n$ vertices and exactly $4n-8$ edges, have been completely characterized,…

The Erd\H{o}s-S\'{o}s Conjecture states that every graph with average degree more than $k-2$ contains all trees of order $k$ as subgraphs. In this paper, we consider a variation of the above conjecture: studying the maximum size of an…

For an integer $k \ge 1$, a (distance) $k$-dominating set of a connected graph $G$ is a set $S$ of vertices of $G$ such that every vertex of $V(G) \setminus S$ is at distance at most~$k$ from some vertex of $S$. The $k$-domination number,…

We show that for each $k\geq 4$ and $n>r\geq k+1$, every $n$-vertex $r$-uniform hypergraph with no Berge cycle of length at least $k$ has at most $\frac{(k-1)(n-1)}{r}$ edges. The bound is exact, and we describe the extremal hypergraphs.…

We consider the case in which mixed graphs (with both directed and undirected edges) are Cayley graphs of Abelian groups. In this case, some Moore bounds were derived for the maximum number of vertices that such graphs can attain. We first…

We suggest two related conjectures dealing with the existence of spanning irregular subgraphs of graphs. The first asserts that any $d$-regular graph on $n$ vertices contains a spanning subgraph in which the number of vertices of each…

The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In: Towards a…

In this paper, motivated by a problem of Scott and a conjecture of Lee, Loh and Sudakov we consider bisections of directed graphs. We prove that every directed graph with $m$ arcs and minimum semidegree at least $d$ admits a bisection in…

We consider the following fundamental realization problem of directed graphs. Given a sequence $S:={a_1 \choose b_1},\dots,{a_n \choose b_n}$ with $a_i,b_i\in \mathbb{Z}_0^+.$ Does there exist a digraph (no loops and no parallel arcs are…

Computing the directed path-width of a directed graph is an NP-hard problem. Even for digraphs of maximum semi-degree 3 the problem remains hard. We propose a decomposition of an input digraph G=(V,A) by a number k of sequences with entries…

A vertex-girth-regular $vgr(v,k,g,\lambda)$-graph is a $k$-regular graph of girth $g$ and order $v$ in which every vertex belongs to exactly $\lambda$ cycles of length $g$. While all vertex-transitive graphs are necessarily…

For their famous algorithm for the disjoint paths problem, Robertson and Seymour proved that there is a function $f$ such that if the tree-width of a graph $G$ with $k$ pairs of terminals is at least $f(k)$, then $G$ contains a…

An orientation of an undirected graph $G$ is an assignment of exactly one direction to each edge of $G$. The oriented diameter of a graph $G$ is the smallest diameter among all the orientations of $G$. The maximum oriented diameter of a…