Related papers: Spanning Factorizations in Vertex-Transitive Digra…
Not every directed acyclic graph (DAG) whose underlying undirected graph is planar admits an upward planar drawing. We are interested in pushing the notion of upward drawings beyond planarity by considering upward $k$-planar drawings of…
This note summarizes the state of what is known about the tractability of the problem ModPath, which asks if an input undirected graph contains a simple st-path whose length satisfies modulo constraints. We also consider the problem…
We introduce the class of interval $H$-graphs, which is the generalization of interval graphs, particularly interval bigraphs. For a fixed graph $H$ with vertices $a_1,a_2,\dots,a_k$, we say that an input graph $G$ with given partition…
Consider a sequence of LexBFS vertex orderings {\sigma}1, {\sigma}2, . . . where each ordering {\sigma}i is used to break ties for {\sigma}i+1. Since the total number of vertex orderings of a finite graph is finite, this sequence must end…
In 1985, Mader conjectured the existence of a function $f$ such that every digraph with minimum out-degree at least $f(k)$ contains a subdivision of the transitive tournament of order $k$. This conjecture is still completely open, as the…
A Schreier decoration is a combinatorial coding of an action of the free group $F_d$ on the vertex set of a $2d$-regular graph. We investigate whether a Schreier decoration exists on various countably infinite transitive graphs as a factor…
A string graph is the intersection graph of curves in the plane. Kratochv\'il previously showed the existence of infinitely many obstacles: graphs that are not string graphs but for which any edge contraction or vertex deletion produces a…
A digraph is {\bf \( k \)-linked} if for arbitary two disjoint vertex sets \(\{s_1, \ldots, s_k\}\) and \(\{t_1, \ldots, t_k\}\), there exist vertex-disjoint directed paths \(P_1, \ldots, P_k\) {such that \(P_i\) is a directed path from…
Let $G$ be a graph and $F:V(G)\to2^N$ be a set function. The graph $G$ is said to be \emph{F-avoiding} if there exists an orientation $O$ of $G$ such that $d^+_O(v)\notin F(v)$ for every $v\in V(G)$, where $d^+_O(v)$ denotes the out-degree…
Let $ H $ be a multi-digraph on $ h $ vertices with $ q $ arcs. An \textbf{$H$-subdivision} in a digraph $D$ is a subdigraph obtained by replacing every arc $uv$ of $H$ with a path from $u$ to $v$ in $D$ such that these paths are pairwise…
The {\sc Directed Maximum Leaf Out-Branching} problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we improve known parameterized algorithms and…
In this paper we present a complete characterization of the smallest sets that block all the simple spanning trees (SSTs) in a complete geometric graph. We also show that if a subgraph is a blocker for all SSTs of diameter at most 4, then…
For a graph (undirected, directed, or mixed), a cycle-factor is a collection of vertex-disjoint cycles covering the entire vertex set. Cycle-factors subject to parity constraints arise naturally in the study of structural graph theory and…
This continues the investigation of a combinatorial model for the variation of dynamics in the family of rational maps of degree two, by concentrating on those varieties in which one critical point is periodic. We prove some general results…
Given a signing $\sigma\colon E(K_{n,n})\to\{-1,+1\}$ of the complete bipartite graph, when does $K_{n,n}$ admit a $1$-factorization in which every perfect matching has discrepancy bounded below by a positive absolute constant? Unlike the…
The first part of the paper studies star-cycle factors of graphs. It characterizes star-cycle factors of a graph $G$ and proves upper bounds for the minimum number of $K_{1,2}$-components in a $\{K_{1,1}, K_{1,2}, C_n\colon n\ge 3\}$-factor…
We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an {\bf out-colouring}. The problem of deciding whether a given digraph has an out-colouring with only two colours (called a…
A recent result by Kardo\v{s}, M\'a\v{c}ajov\'a and Zerafa [J. Comb. Theory, Ser. B. 160 (2023) 1--14] related to the famous Berge-Fulkerson conjecture implies that given an arbitrary set of odd pairwise edge-disjoint cycles, say $\mathcal…
A digraph $G$ is \emph{$k$-geodetic} if for any (not necessarily distinct) vertices $u,v$ there is at most one directed walk from $u$ to $v$ with length not exceeding $k$. The order of a $k$-geodetic digraph with minimum out-degree $d$ is…
We call an oriented odd cycle alternating if it has exactly one vertex whose in-degree and out-degree are both positive. In this paper, we investigate whether certain graphs admit an orientation that avoids alternating odd cycles as…