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A simple connected graph G is called a p-quasi k-cyclic graph, if there exists a subset S of vertices such that |S|=p, G-S is k-cyclic and there is no a subset S` of V(G) such that |S`|<|S| and G-S` is k-cyclic. The aim of this paper is to…
An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and…
We exhibit non-switching-isomorphic signed graphs that share a common underlying graph and common chromatic polynomials, thereby answering a question posed by Zaslavsky. For various joins of all-positive or all-negative signed complete…
Recurrence is a fundamental characteristic of dynamical systems with complicated behavior. Understanding the inner structure of recurrence is challenging, especially if the system has many degrees of freedom and is subject to noise. We…
Let $G=(V,E)$ be a simple graph with $n = |V|$ vertices and $m = |E|$ edges. The first and second Zagreb indices are among the oldest and the most famous topological indices, defined as $M_1 = \sum_{i \in V} d_i^2$ and $M_2 = \sum_{(i, j)…
We study the chromatic number of the curve graph of a surface. We show that the chromatic number grows like k log k for the graph of separating curves on a surface of Euler characteristic -k. We also show that the graph of curves that…
A signed graph $(G, \sigma)$ is a graph $G$ along with a function $\sigma: E(G) \to \{+,-\}$. A closed walk of a signed graph is positive (resp., negative) if it has an even (resp., odd) number of negative edges, counting repetitions. A…
We prove that the number of Hamiltonian paths on the complement of an acyclic digraph is equal to the number of cycle covers. As an application, we obtain a new expansion of the chromatic symmetric function of incomparability graphs in…
Topological phases of matter have sparked an immense amount of activity in recent decades. Topological materials are classified by topological invariants that act as a non-local order parameter for any symmetry and condition. As a result,…
Topological photonics provides a powerful framework to describe and understand many nontrivial wave phenomena in complex electromagnetic platforms. The topological index of a physical system is an abstract global property that depends on…
An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and…
Let G = (V, E) be a finite simple connected graph. We say a graph G realizes a code of the type 0^s_1 1^t_1 0^s_2 1^t_2 ... 0^s_k1^t_k if and only if G can obtained from the code by some rule. Some classes of graphs such as threshold and…
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…
In 1975 Erd\H{o}s initiated the study of the following very natural question. What can be said about the chromatic number of unit distance graphs in $\mathbb{R}^2$ that have large girth? Over the years this question and its natural…
Let $k, d$ ($2d \leq k)$ be two positive integers. We generalize the well studied notions of $(k,d)$-colorings and of the circular chromatic number $\chi_c$ to signed graphs. This implies a new notion of colorings of signed graphs, and the…
We study combinatorial indicators related to the characteristic phase transitions associated with coloring a graph optimally and finding a maximum independent set. In particular, we investigate the role of the acyclic orientations of the…
The abstract chromatic number was introduced by Razborov and Coregliano in 2020 in using the language of model theory, and was used to extend the Erd\H os-Stone-Simonovits theorem to graphs with extra structures. A purely combinatorial…
We investigate a notion of $\times$-homotopy of graph maps that is based on the internal hom associated to the categorical product in the category of graphs. It is shown that graph $\times$-homotopy is characterized by the topological…
A \textit{$k$-total coloring} of a graph $G$ is an assignment of $k$ colors to its elements (vertices and edges) so that adjacent or incident elements have different colors. The total chromatic number is the smallest integer $k$ for which…
We define the fundamental quandle of a spatial graph and several invariants derived from it. In the category of graph tangles, we define an invariant based on the walks in the graph and cocycles from nonabelian quandle cohomology.