Related papers: Two cycle-chord graphs are $e$-positive

A chord of a cycle $C$ is an edge joining two non-consecutive vertices of $C$. A cycle $C$ in a graph $G$ is chorded if the vertex set of $C$ induces at least one chord. In this paper, we prove that if $G$ is a graph with order $n\geq 6$…

In this work we investigate the chordality of squares and line graph squares of graphs. We prove a sufficient condition for the chordality of squares of graphs not containing induced cycles of length at least five. Moreover, we characterize…

We relate matroid connectivity to Tutte-connectivity in an infinite graph. Moreover, we show that the two cycle matroids, the finite-cycle matroid and the cycle matroid, in which also infinite cycles are taken into account, have the same…

The periodic discrete Toda equation defined over finite fields has been studied. We obtained the finite graph structures constructed by the network of states where edges denote possible time evolutions. We simplify the graphs by introducing…

In 1990, Hendry conjectured that all Hamiltonian chordal graphs are cycle extendable. After a series of papers confirming the conjecture for a number of graph classes, the conjecture is yet refuted by Lafond and Seamone in 2015. Given that…

Asymptotic expansions of Gaussian integrals may often be interpreted as generating functions for certain combinatorial objects (graphs with additional data). In this article we discuss a general approach to all such cases using colored…

How many edges in an $n$-vertex graph will force the existence of a cycle with as many chords as it has vertices? Almost 30 years ago, Chen, Erd\H{o}s and Staton considered this question and showed that any $n$-vertex graph with $2n^{3/2}$…

We show that every edge in a 2-edge-connected planar cubic graph is either contained in a 2-edge-cut or is a chord of some cycle that is contained in a 2-factor of the graph. As a consequence, we show that every edge in a cyclically…

A cycle $C$ of length $k$ in graph $G$ is extendable if there is another cycle $C'$ in $G$ with $V(C) \subset V(C')$ and length $k+1$. A graph is cycle extendable if every non-Hamiltonian cycle is extendable. In 1990 Hendry conjectured that…

In this note we propose an $\omega$-operadical way to prove the existence of the $\omega$-graph of the $\omega$-graphs and the reflexive $\omega$- graph of the reflexive $\omega$-graphs.

A graph $X$ is 2-spanning cyclable if for any pair of distinct vertices $u$ and $v$ there is a 2-factor of $X$ consisting of two cycles such that $u$ and $v$ belong to distinct cycles. In this paper we examine the 2-spanning cyclability of…

Given a graph $G$ and a subset $X$ of vertices of $G$ with size at least two, we denote by $N^2_G(X)$ the set of vertices of $G$ that have at least two neighbors in $X$. We say that a bipartite graph $G$ with sides $A$ and $B$ satisfies the…

Given two graphs, a mapping between their edge-sets is cycle-continuous, if the preimage of every cycle is a cycle. The motivation for this notion is Jaeger's conjecture that for every bridgeless graph there is a cycle-continuous mapping to…

The \emph{Antimagic Graph Conjecture} asserts that every connected graph $G = (V, E)$ except $K_2$ admits an edge labeling such that each label $1, 2, ..., |E|$ is used exactly once and the sums of the labels on all edges incident with a…

Let $G$ be a simple graph of order $n$ with eigenvalues $\lambda_1(G)\geq \cdots \geq \lambda_n(G)$. Define \[s^+(G)=\sum_{\lambda_i >0} \lambda_i^2(G), \quad s^-(G)=\sum_{\lambda_i<0} \lambda_i^2(G).\] It was conjectured by Elphick,…

A long-standing conjecture of Thomassen says that every longest cycle of a $3$-connected graph has a chord. Thomassen (2018) proved that if $G$ is a $2$-connected cubic graph, then any longest cycle must have a chord. He also showed that in…

Let $L(G)$ be the set of all subgroups of a group $G$. The subgroup generating bipartite graph $\mathcal{B}(G)$ defined on $G$ is a bipartite graph whose vertex set is the union of two sets $G \times G$ and $L(G)$, and two vertices $(a, b)…

In this paper, we give a lengthy proof of a small result! A graph is bisplit if its vertex set can be partitioned into three stable sets with two of them inducing a complete bipartite graph. We prove that these graphs satisfy the…

We conjecture that a 2-connected graph $G$ of order $n$, in which $d(x)+d(y)\geq n-k$ for every pair of non-adjacent vertices $x$ and $y$, contains a cycle of length $n-k$ ($k<n/2$), unless $G$ is bipartite and $n-k$ is odd. This…

It is well known that the edge vector space of an oriented graph can be decomposed in terms of cycles and cocycles (alias cuts, or bonds), and that a basis for the cycle and the cocycle spaces can be generated by adding and removing edges…