Related papers: A note on antimagic orientations of even regular g…
In 2003, Bohman, Frieze, and Martin initiated the study of randomly perturbed graphs and digraphs. For digraphs, they showed that for every $\alpha>0$, there exists a constant $C$ such that for every $n$-vertex digraph of minimum…
We consider non-trivial homomorphisms to reflexive oriented graphs in which some pair of adjacent vertices have the same image. Using a notion of convexity for oriented graphs, we study those oriented graphs that do not admit such…
Let ${\mathcal D}_{n,d}$ be the set of all $d$-regular directed graphs on $n$ vertices. Let $G$ be a graph chosen uniformly at random from ${\mathcal D}_{n,d}$ and $M$ be its adjacency matrix. We show that $M$ is invertible with probability…
In 1974, Erd\H{o}s asked the following question: given a graph $G$ and a directed graph $\vec{H}$, how many ways are there to orient the edges of $G$ such that it does not contain $\vec{H}$ as a subgraph? We denote this value by $D(G,…
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…
Let D be a directed graph with vertex set V and order n. An anti-directed hamiltonian cycle H in D is a hamiltonian cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. An anti-directed…
In this paper, we study oriented bipartite graphs. In particular, we introduce "bitransitive" graphs. Several characterizations of bitransitive bitournaments are obtained. We show that bitransitive bitounaments are equivalent to acyclic…
Graph labellings have been a very fruitful area of research in the last four decades. However, despite the staggering number of papers published in the field (over 1000), few general results are available, and most papers deal with…
A graph $G = (V, E)$ of order $p$ and size $q$ is said to be local antimagic if there exists a bijection $g:E(G) \to \{1,2,\ldots,q\}$ such that for any pair of adjacent vertices $u$ and $v$, $g^+(u)\ne g^+(v)$, where $g^+(u)=\sum_{uv\in…
Lov\'{a}sz conjectured that every connected vertex-transitive graph contains a hamilton path in 1970. First we reveal the structure of connected vertex-transitive graphs with an odd number of vertices. Then we prove that every connected…
The well-known 1-2-3 Conjecture asserts that the edges of every graph without an isolated edge can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every…
Graph labeling is a technique that assigns unique labels or weights to the vertices or edges of a graph, often used to analyze and solve various graph-related problems. There are few methods with certain limitations conducted by researchers…
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…
A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph…
Kriesel conjectured that every minimally $1$-tough graph has a vertex with degree precisely $2$. Katona and Varga (2018) proposed a generalized version of this conjecture which says that every minimally $t$-tough graph has a vertex with…
We initiate the study of $k$-edge-connected orientations of undirected graphs through edge flips for $k \geq 2$. We prove that in every orientation of an undirected $2k$-edge-connected graph, there exists a sequence of edges such that…
In 1999, De Simone and K\"{o}rner conjectured that every graph without induced $C_5,C_7,\overline{C}_7$ contains a clique cover $\mathcal C$ and a stable set cover $\mathcal I$ such that every clique in $\mathcal C$ and every stable set in…
Since Sedl\'{a}$\breve{\mbox{c}}$ek introduced the notion of magic labeling of a graph in 1963, a variety of magic labelings of a graph have been defined and studied. In this paper, we study consecutive edge magic labelings of a connected…
Equistable graphs are graphs admitting positive weights on vertices such that a subset of vertices is a maximal stable set if and only if it is of total weight $1$. In $1994$, Mahadev et al.~introduced a subclass of equistable graphs,…
Erd{\H o}s (1963) initiated extensive graph discrepancy research on 2-edge-colored graphs. Gishboliner, Krivelevich, and Michaeli (2023) launched similar research on oriented graphs. They conjectured the following extension of Dirac's…