Related papers: Antimagic orientations of even regular graphs

Let $n$ be sufficiently large and suppose that $G$ is a digraph on $n$ vertices where every vertex has in- and outdegree at least $n/2$. We show that $G$ contains every orientation of a Hamilton cycle except, possibly, the antidirected one.…

The incidence matrix of a graph is totally unimodular if and only if the graph is bipartite, i.e., it contains no odd cycles. We extend the characterization of total unimodularity to hypergraphs whose hyperedges of size at least four are…

Let $D$ be a digraph. A subset $S$ of $V(D)$ is a stable set if every pair of vertices in $S$ is non-adjacent in $D$. A collection of disjoint paths $\mathcal{P}$ of $D$ is a path partition of $V(D)$, if every vertex in $V(D)$ is on a path…

Let $D$ be a digraph. A collection of disjoint sets of vertices (respec., collection of disjoint subdigraphs) $\mathcal{H}$ of $D$ and a vertex subset (or subdigraph) $Q$ of $D$ are orthogonal if every set (respec., subdigraph) $H \in…

In a digraph $D=(V,A)$, an oriented path is a sequence $P=x_1x_2\dots x_p$ of distinct vertices such that either $x_ix_{i+1}\in A$ or $x_{i+1}x_{i}\in A$ or both for every $i\in [p-1]$. If $x_ix_{i+1}\in A$ in $P$, then $x_ix_{i+1}$ is a…

For an oriented graph $D$ and a set $X\subseteq V(D)$, the inversion of $X$ in $D$ is the digraph obtained by reversing the orientations of the edges of $D$ with both endpoints in $X$. The inversion number of $D$, $\textrm{inv}(D)$, is the…

We introduce the concept of link-irregular labelings for graphs, extending the notion of link-irregular graphs through edge labeling with positive integers. A labeling is link-irregular if every vertex has a uniquely labeled subgraph…

We settle affirmatively a conjecture posed in [S. M. Hegde, Set colorings of graphs, European Journal of Combinatorics 30 (4) (2009), 986--995]: If some subsets of a set X are assigned injectively to all vertices of a complete bipartite…

An oriented graph is a directed graph without directed 2-cycles. Poljak and Turz\'{i}k (1986) proved that every connected oriented graph $G$ on $n$ vertices and $m$ arcs contains an acyclic subgraph with at least $\frac{m}{2}+\frac{n-1}{4}$…

A signed graph is a graph with a function that assigns a label of positive or negative to each edge. The sign of a circle is the product of the signs of its edges; a graph is balanced if all of its circles are positive. A set of edges whose…

In graph theory, a graceful labeling of a graph with m edges is a labeling of its vertices with a subset of the integers ranging from 0 to m inclusive, such that no two vertices share a label, and each edge is uniquely identified by the…

If $A$ is a finite Abelian group, then a labeling $f \colon E (G) \rightarrow A$ of the edges of some graph $G$ induces a vertex labeling on $G$; the vertex $u$ receives the label $\sum_{v\in N(u)}f (v)$, where $N(u)$ is an open…

We study graphs which admit an acyclic orientation that contains an out-branching and in-branching which are arc-disjoint (such an orientation is called {\bf good}). A {\bf 2T-graph} is a graph whose edge set can be decomposed into two…

We prove that for every oriented graph $D$ and every choice of positive integers $k$ and $\ell$, there exists an oriented graph $D^*$ along with a surjective homomorphism $\psi\colon V(D^*) \to V(D)$ such that: (i) girth$(D^*) \geq\ell$;…

The study of graph discrepancy problems, initiated by Erd\H{o}s in the 1960s, has received renewed attention in recent years. In general, given a $2$-edge-coloured graph $G$, one is interested in embedding a copy of a graph $H$ in $G$ with…

Nash-Williams' Strong Immersion Conjecture states that graphs are well-quasi-ordered by the strong immersion relation. That is, given infinitely many graphs, one graph contains another graph as a strong immersion. In this paper we study the…

An oriented graph is a directed graph without any cycle of length at most 2. To push a vertex of a directed graph is to reverse the orientation of the arcs incident to that vertex. Klostermeyer and MacGillivray defined push graphs which are…

\qquad A \emph{coloring} of a digraph $D=(V,E)$ is a coloring of its vertices following the rule: Let $uv$ be an arc in $D$. If the tail $u$ is colored first, then the head $v$ should receive a color different from that of $u$. The…

A digraph $H$ is called ubiquitous if every digraph that contains arbitrarily many vertex-disjoint copies of $H$ also contains infinitely many vertex-disjoint copies of $H$. We study oriented double rays, that is, digraphs $H$ whose…

Let $G$ be a $d$-regular graph and let $F\subseteq\{0, 1, 2, \ldots, d\}$ be a list of forbidden out-degrees. Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati conjectured that if $|F|<\tfrac{1}{2}d$, then $G$ should admit an $F$-avoiding…