Related papers: Disconnecting strongly regular graphs

Let $G$ be a nontrivial connected and edge-colored graph. An edge-cut $R$ of $G$ is called a rainbow cut if no two edges of it are colored with a same color. An edge-colored graph $G$ is called rainbow disconnected if for every two distinct…

Given a set $\Sg$ of bipartitions of some finite set $X$ of cardinality at least 2, one can associate to $\Sg$ a canonical $X$-labeled graph $\B(\Sg)$, called the Buneman graph. This graph has several interesting mathematical properties -…

Let $G$ be a (multi)graph of order $n$ and let $u,v$ be vertices of $G$. The maximum number of internally disjoint $u$-$v$ paths in $G$ is denoted by $\kappa_G(u,v)$, and the maximum number of edge-disjoint $u$-$v$ paths in $G$ is denoted…

A path in an edge-colored graph $G$ is rainbow if no two edges of it are colored the same. The graph $G$ is rainbow-connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair…

In this paper, we show that every $O(m)$-edge-connected simple graph $G$ of size divisible by $m$ with minimum degree at least $2^{O(m)}$ has an edge-decomposition into isomorphic copies of any given tree $T$ of size $m$. Moreover, the…

A non-complete geometric distance-regular graph is the point graph of a partial geometry in which the set of lines is a set of Delsarte cliques. In this paper, we prove that for fixed integer $m\geq 2$, there are only finitely many…

For a set of five edges, a graph splits if one of the associated Dodgson polynomials is equal to zero. A graph G splitting for every set of five edges is a minor-closed property. As such there is a finite set of forbidden minors F such that…

A graph $G$ is $k$-degenerate if it can be transformed into an empty graph by subsequent removals of vertices of degree $k$ or less. We prove that every connected planar graph with average degree $d \ge 2$ has a 4-degenerate induced…

We prove that for a connected simple graph $G$ with $n\le 10$ vertices, and two longest paths $C$ and $D$ in $G$, the intersection of vertex sets $V(C)\cap V(D)$ is a separator. This shows that the graph found previously with $n=11$, in…

The simplex graph $S(G)$ of a graph $G$ is defined as the graph whose vertices are the cliques of $G$ (including the empty set), with two vertices being adjacent if, as cliques of $G$, they differ in exactly one vertex. Simplex graphs form…

The purpose of this paper is to characterize graphs that do not have a large $K_{2,n}$-minor. As corollaries, it is proved that, for any given positive integer $n$, every sufficiently large 3-connected graph with minimum degree at least…

The skewness of a graph G is the minimum number of edges in G whose removal results in a planar graph. By appropriately introducing a weight to each edge of a graph, we determine, among other thing, the skewness of the generalized Petersen…

Let $\mu_2(G)$ be the second smallest Laplacian eigenvalue of a graph $G$. The vertex connectivity of $G$, written $\kappa(G)$, is the minimum size of a vertex set $S$ such that $G-S$ is disconnected. Fiedler proved that $\mu_2(G) \le…

Motivated by work of Haythorpe, Thomassen and the author showed that there exists a positive constant $c$ such that there is an infinite family of 4-regular 4-connected graphs, each containing exactly $c$ hamiltonian cycles. We complement…

This article investigates the connectivity dimension of a graph. We introduce this concept in analogy to the metric dimension of a graph, providing a graph parameter that measures the heterogeneity of the connectivity structure of a graph.…

A graph $G$ is a non-separating planar graph if there is a drawing $D$ of $G$ on the plane such that (1) no two edges cross each other in $D$ and (2) for any cycle $C$ in $D$, any two vertices not in $C$ are on the same side of $C$ in $D$.…

The line graph of a graph $G$ is the graph $L(G)$ whose vertex set is the edge set of $G$ and there is an edge between $e,f\in E(G)$ if $e$ and $f$ share an endpoint in $G$. A graph is called line graph if it is a line graph of some graph.…

An edge $e$ of a graph $G$ is called deletable for some orientation $o$ if the restriction of $o$ to $G-e$ is a strong orientation. Inspired by a problem of Frank, in 2021 H\"orsch and Szigeti proposed a new parameter for $3$-edge-connected…

The visibility graph of a finite set of points in the plane has the points as vertices and an edge between two vertices if the line segment between them contains no other points. This paper establishes bounds on the edge- and…

We pursue the study of edge-irregulators of graphs, which were recently introduced in [Fioravantes et al. Parametrised Distance to Local Irregularity. IPEC, 2024]. That is, we are interested in the parameter Ie(G), which, for a given graph…