Related papers: Graph Cordiality -- Extremes and Preservers

An antimagic labeling a connected graph $G$ is a bijection from the set of edges $E(G)$ to $\{1,2,\dots,|E(G)|\}$ such that all vertex sums are pairwise distinct, where the vertex sum at vertex $v$ is the sum of the labels assigned to edges…

For a graph $G,$ we denote the number of connected subgraphs of $G$ by $F(G)$. For a tree $T$, $F(T)$ has been studied extensively and it has been observed that $F(T)$ has a reverse correlation with Wiener index of $T$. Based on that, we…

The median of a graph $G$ with weighted vertices is the set of all vertices $x$ minimizing the sum of weighted distances from $x$ to the vertices of $G$. For any integer $p\ge 2$, we characterize the graphs in which, with respect to any…

A graph is said to be well-edge-dominated if all its minimal edge dominating sets are minimum. It is known that every well-edge-dominated graph $G$ is also equimatchable, meaning that every maximal matching in $G$ is maximum. In this paper,…

A (Euclidean) greedy drawing of a graph is a drawing in which, for any two vertices $s,t$ ($s \neq t$), there is a neighbor vertex of $s$ that is closer to $t$ than to $s$ in the Euclidean distance. Greedy drawings are important in the…

A graph $H$ is an \emph{isometric} subgraph of $G$ if $d_H(u,v)= d_G(u,v)$, for every pair~$u,v\in V(H)$. A graph is \emph{distance preserving} if it has an isometric subgraph of every possible order. A graph is \emph{sequentially distance…

For a graph $G = (V, E)$, the $\gamma$-graph of $G$ is the graph whose vertex set is the collection of minimum dominating sets, or $\gamma$-sets of $G$, and two $\gamma$-sets are adjacent if they differ by a single vertex and the two…

We classify the connected-homogeneous digraphs with more than one end. We further show that if their underlying undirected graph is not connected-homogeneous, they are highly-arc-transitive.

To push a vertex $v$ of a directed graph $\overrightarrow{G}$ is to change the orientations of all the arcs incident with $v$. An oriented graph is a directed graph without any cycle of length at most 2. An oriented clique is an oriented…

We present the first comprehensive analysis of temporal settings for directed temporal graphs, fully resolving their hierarchy with respect to support, reachability, and induced-reachability equivalence. These notions, introduced by…

Among all uniform hypergraphs with even uniformity, the odd-transversal or odd-bipartite hypergraphs are more close to bipartite simple graphs from the viewpoint of both structure and spectrum. A hypergraph is called minimal…

Orienting the edges of an undirected graph such that the resulting digraph satisfies some given constraints is a classical problem in graph theory, with multiple algorithmic applications. In particular, an $st$-orientation orients each edge…

A sequence $D=(d_1,d_2,\ldots,d_n)$ of non-negative integers is called a graphic sequence if there is a simple graph with vertices $v_1,v_2,\ldots,v_n$ such that the degree of $v_i$ is $d_i$ for $1\leq i\leq n$. Given a graph theoretical…

A labeling of a digraph $D$ with $m$ arcs is a bijection from the set of arcs of $D$ to $\{1, \ldots, m\}$. A labeling of $D$ is antimagic if no two vertices in $D$ have the same vertex-sum, where the vertex-sum of a vertex $u\in V(D)$ for…

A graph $X$ is said to be "unstable" if the direct product $X \times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is "nontrivially unstable" if…

A plane graph is said to be a rectangular graph if each of its edges can be oriented horizontal or vertical, its internal regions are four-sided and it has a rectangular enclosure. If dual of a planar graph is a rectangular graph, then the…

A majority coloring of an undirected graph is a vertex coloring in which for each vertex there are at least as many bi-chromatic edges containing that vertex as monochromatic ones. It is known that for every countable graph a majority…

A graph is distance magic if it admits a bijective labeling of its vertices by integers from $1$ up to the order of the graph in such a way that the sum of the labels of all the neighbors of a vertex is independent of a given vertex. We…

A retract of a graph $\Gamma$ is an induced subgraph $\Psi$ of $\Gamma$ such that there exists a homomorphism from $\Gamma$ to $\Psi$ whose restriction to $\Psi$ is the identity map. A graph is a core if it has no nontrivial retracts. In…

Given a finite group $G$, the invariably generating graph of $G$ is defined as the undirected graph in which the vertices are the nontrivial conjugacy classes of $G$, and two classes are adjacent if and only if they invariably generate $G$.…