Related papers: New results on word-representable graphs
The class of Parikh word representable graphs were recently introduced. In this work, we further develop its general theory beyond the binary alphabet. Our main result shows that this class is equivalent to the class of bipartite…
If the vertices of a graph $G$ are colored with $k$ colors such that no adjacent vertices receive the same color and the sizes of any two color classes differ by at most one, then $G$ is said to be equitably $k$-colorable. Let $|G|$ denote…
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 introduce a variation of interval graphs, called veto interval (VI) graphs. A VI graph is represented by a set of closed intervals, each containing a point called a veto mark. The edge $ab$ is in the graph if the intervals corresponding…
Given a graph $G$ with $n$ vertices and a bijective labeling of the vertices using the integers $1,2,\ldots, n$, we say $G$ has a peak at vertex $v$ if the degree of $v$ is greater than or equal to 2, and if the label on $v$ is larger than…
A graph is vertex-transitive if its automorphism group acts transitively on vertices of the graph. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper,…
A graph is apex if it can be made planar by deleting a vertex, that is, $\exists v$ such that $G-v$ is planar. We define the related notions of edge apex, $\exists e$ such that $G-e$ is planar, and contraction apex, $\exists e$ such that…
Given a set D of nonnegative integers, we derive the asymptotic number of graphs with a givenvnumber of vertices, edges, and such that the degree of every vertex is in D. This generalizes existing results, such as the enumeration of graphs…
A graph $G=(V,E)$ is called a pairwise compatibility graph (PCG) if there exists an edge-weighted tree $T$ and two non-negative real numbers $d_{min}$ and $d_{max}$ such that each leaf $u$ of $T$ corresponds to a vertex $u \in V$ and there…
Topological drawings are natural representations of graphs in the plane, where vertices are represented by points, and edges by curves connecting the points. Topological drawings of complete graphs and of complete bipartite graphs have been…
A CIS graph is a graph in which every maximal stable set and every maximal clique intersect. A graph is well-covered if all its maximal stable sets are of the same size, co-well-covered if its complement is well-covered, and…
A graph $G$ is \emph{$(a,b)$-choosable} if given any list assignment $L$ with $|L(v)|=a$ for each $v\in V(G)$ there exists a function $\varphi$ such that $\varphi(v)\in L(v)$ and $|\varphi(v)|=b$ for all $v\in V(G)$, and whenever vertices…
Deciding whether a planar graph (even of maximum degree $4$) is $3$-colorable is NP-complete. Determining subclasses of planar graphs being $3$-colorable has a long history, but since Gr\"{o}tzsch's result that triangle-free planar graphs…
Inspired by distributed algorithms, we introduce a new class of finite graph automata that recognize precisely the graph languages definable in monadic second-order logic. For the cases of words and trees, it has been long known that the…
Let $G = (V, E)$ be a graph and $\lambda $ a non-negative integer. A graph $G$ is called a $(\lambda, 1)$-{\em graph} if $ (c0)$ $G$ is neither a complete graph no an edge-empty graph, $ (c1)$ every edge in $G$ belongs to exactly $\lambda$…
A matching of a graph is a set of edges without common end vertex. A graph is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. Recently, Biedl and Wittnebel proved that every 1-planar graph…
Jamison and Sprague defined a graph $G$ to be a $k$-threshold graph with thresholds $\theta_1 , \ldots, \theta_k$ (strictly increasing) if one can assign real numbers $(r_v)_{v \in V(G)}$, called ranks, such that for every pair of vertices…
Generalizing the notion of staircase words, introduced by Knopfmacher et.\ al, we define staircase graph words. These are functions $w$ from the vertex set $V$ of a graph into the set $\{1,2,\ldots,k\}$, such that $|w(x)-w(y)|\leq 1$, for…
A graph G = (V,E) is called fully regular if for every independent set $I\subset V$ , the number of vertices in $V\setminus$ I that are not connected to any element of I depends only on the size of I. A linear ordering of the vertices of G…
Exchangeable random graphs serve as an important probabilistic framework for the statistical analysis of network data. In this work we develop an alternative parameterization for a large class of exchangeable random graphs, where the nodes…