Related papers: Removable edges in cubic matching covered graphs

Given a graph $G$, two edges $e_{1},e_{2}\in E(G)$ are said to have a common edge $e$ if $e$ joins an endvertex of $e_{1}$ to an endvertex of $e_{2}$. A subset $B\subseteq E(G)$ is an edge open packing set in $G$ if no two edges of $B$ have…

In this paper, we explain the regularity, projective dimension and depth of edge ideal of some classes of graphs in terms of invariants of graphs. We show that for a $C_5$-free vertex decomposable graph $G$, $\T{reg}(R/I(G))= c_G$, where…

A graph $G=(V,E)$ is a $k$-critical graph if $G$ is not $(k -1)$-colorable but $G-e$ is $(k-1)$-colorable for every $e\in E(G)$. In this paper, we construct a family of 4-critical planar graphs with $n$ vertices and $\frac{7n-13}{3}$ edges.…

The (Perfect) Matching Cut problem is to decide if a connected graph has a (perfect) matching that is also an edge cut. The Disconnected Perfect Matching problem is to decide if a connected graph has a perfect matching that contains a…

Contraction of triangles is a standard operation in the study of cubic graphs, as it reduces the order of the graph while typically preserving many of its properties. In this paper, we investigate the converse problem, wherein certain…

In Partition Into Complementary Subgraphs (Comp-Sub) we are given a graph $G=(V,E)$, and an edge set property $\Pi$, and asked whether $G$ can be decomposed into two graphs, $H$ and its complement $\overline{H}$, for some graph $H$, in such…

A normal odd partition T of the edges of a cubic graph is a partition into trails of odd length (no repeated edge) such that each vertex is the end vertex of exactly one trail of the partition and internal in some trail. For each vertex v,…

Let $G$ and $H$ be simple 3-connected graphs such that $G$ has an $H$-minor. An edge $e$ in $G$ is called {\it $H$-deletable} if $G\backslash e$ is 3-connected and has an $H$-minor. The main result in this paper establishes that, if $G$ has…

The vertex (resp. edge) metric dimension of a connected graph G; denoted by dim(G) (resp. edim(G)), is defined as the size of a smallest set S in V (G) which distinguishes all pairs of vertices (resp. edges) in G: Bounds dim(G) <=…

Let G be a cubic graph, with girth at least five, such that for every partition X,Y of its vertex set with |X|,|Y|>6 there are at least six edges between X and Y. We prove that if there is no homeomorphic embedding of the Petersen graph in…

We study noncrossing geometric graphs and their disjoint compatible geometric matchings. Given a cycle (a polygon) P we want to draw a set of pairwise disjoint straight-line edges with endpoints on the vertices of P such that these new…

We show that every cubic bridgeless graph with n vertices has at least 3n/4-10 perfect matchings. This is the first bound that differs by more than a constant from the maximal dimension of the perfect matching polytope.

For $k \geq 1$ and a graph $G$ let $\nu_k(G)$ denote the size of a maximum $k$-edge-colorable subgraph of $G$. Mkrtchyan, Petrosyan and Vardanyan proved that $\nu_2(G)\geq \frac45\cdot |V(G)|$, $\nu_3(G)\geq \frac76\cdot |V(G)|$ for any…

A graph $G$ is called matching covered if all of its edges are contained in some perfect matching of $G$. Furthermore, a cycle $C \subseteq G$ is called conformal if $G - V(C)$ has a perfect matching and $G$ itself is called cycle-conformal…

A tree containing exactly two non-pendant vertices is called a double-star. Let $k_1$ and $k_2$ be two positive integers. The double-star with degree sequence $(k_1+1, k_2+1, 1, \ldots, 1)$ is denoted by $S_{k_1, k_2}$. If $G$ is a cubic…

Edge polytopes is a class of interesting polytope with rich algebraic and combinatorial properties, which was introduced by Ohsugi and Hibi. In this papar, we follow a previous study on cutting edge polytopes by Hibi, Li and Zhang. Instead…

A graph is called equimatchable if all of its maximal matchings have the same size. Due to Eiben and Kotrb\v{c}\'{i}k,, any connected graph with odd order and independence number $\alpha(G)$ at most $2$ is equimatchable. Akbari et al.…

A brick is a 3-connected graph such that the graph obtained from it by deleting any two distinct vertices has a perfect matching. A brick is minimal if for every edge e the deletion of e results in a graph that is not a brick. We prove a…

The colouring number col(G) of a graph G is the smallest integer k for which there is an ordering of the vertices of G such that when removing the vertices of G in the specified order no vertex of degree more than k-1 in the remaining graph…

Perfect Matching-Cut is the problem of deciding whether a graph has a perfect matching that contains an edge-cut. We show that this problem is NP-complete for planar graphs with maximum degree four, for planar graphs with girth five, for…