Related papers: Nice vertices in cubic graphs
A {\em brick} is a non-bipartite matching covered graph without non-trivial tight cuts. Bricks are building blocks of matching covered graphs. We say that an edge $e$ in a brick $G$ is {\em $b$-invariant} if $G-e$ is matching covered and a…
Lov\'asz (1987) proved that every matching covered graph $G$ may be uniquely decomposed into a list of bricks (nonbipartite) and braces (bipartite); we let $b(G)$ denote the number of bricks. An edge $e$ is removable if $G-e$ is also…
A graph $G$ with four or more vertices is called bicritical if the removal of any pair of distinct vertices of $G$ results in a graph with a perfect matching. A bicritical graph is minimal if the deletion of each edge results in a…
An edge $e$ of a matching covered graph $G$ is removable if $G-e$ is also matching covered. The notion of removable edge arises in connection with ear decompositions of matching covered graphs introduced by Lov\'asz and Plummer. A…
A conjecture of Berge suggests that every bridgeless cubic graph can have its edges covered with at most five perfect matchings. Since three perfect matchings suffice only when the graph in question is $3$-edge-colourable, the rest of cubic…
A connected graph G is matching covered if every edge lies in some perfect matching of G. Lovasz proved that every matching covered graph G can be uniquely decomposed into a list of bricks (nonbipartite) and braces (bipartite) up to…
A connected graph $G$ with at least two vertices is matching covered if each of its edges lies in a perfect matching. A matching covered graph is minimal if the removal of any edge results in a graph that is no longer matching covered. An…
Let G be a bridgeless cubic graph. A well-known conjecture of Berge and Fulkerson can be stated as follows: there exist five perfect matchings of G such that each edge of G is contained in at least one of them. Here, we prove that in each…
A graph $G$ is well-covered if it has no isolated vertices and all the maximal independent sets have the same cardinality. If furthermore two times this cardinality is equal to $|V(G)|$, the graph $G$ is called very well-covered. The class…
{ An edge $e$ in a matching covered graph $G$ is {\em removable} if $G-e$ is matching covered, which was introduced by Lov\'asz and Plummer in connection with ear decompositions of matching covered graphs. A {\it brick}} is a non-bipartite…
We present evidence in support of a conjecture that a bipartite graph with at least five vertices in each part and |E(G)| \geq 4 |V(G)| - 17 is intrinsically knotted. We prove the conjecture for graphs that have exactly five or exactly six…
A $3$-connected graph $G$ is a brick if, for any two vertices $u$ and $v$, the graph $G-\{u,v\}$ has a perfect matching. Deleting an edge $e$ from a brick $G$ results in a graph with zero, one or two vertices of degree two. The…
Lov{\'a}sz showed that a matching covered graph $G$ has an ear decomposition starting with an arbitrary edge of $G$. Let $G$ be a graph which has a perfect matching. We call $G$ cycle-nice if for each even cycle $C$ of $G$, $G-V(C)$ has a…
It is proved that for $n \geq 6$, the number of perfect matchings in a simple connected cubic graph on $2n$ vertices is at most $4 f_{n-1}$, with $f_n$ being the $n$-th Fibonacci number. The unique extremal graph is characterized as well.…
Motivated by the concept of well-covered graphs, we define a graph to be well-bicovered if every vertex-maximal bipartite subgraph has the same order (which we call the bipartite number). We first give examples of them, compare them with…
Berge Conjecture states that every bridgeless cubic graph has 5 perfect matchings such that each edge is contained in at least one of them. In this paper, we show that Berge Conjecture holds for two classes of cubic graphs, cubic graphs…
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 $G$ is near-bipartite if it has a pair of edges $\alpha$ and $\beta$ such that $G-\{\alpha,\beta\}$…
A connected graph $\G$ is said to be {\it distance-balanced} whenever for any pair of adjacent vertices $u,v$ of $\G$ the number of vertices closer to $u$ than to $v$ is equal to the number of vertices closer to $v$ than to $u$. In…
For distinct vertices $u,v$ in a graph $G$, let $\kappa_G(u,v)$ denote the maximum number of internally disjoint $u$-$v$ paths in $G$. Then, $\kappa_G(u,v) \leq \min\{ \mbox{deg}_G(u), \mbox{deg}_G(v) \}$. If equality is attained for every…
The $k$-token graph $T_k(G)$ is the graph whose vertices are the $k$-subsets of vertices of a graph $G$, with two vertices of $T_k(G)$ adjacent if their symmetric difference is an edge of $G$. We explore when $T_k(G)$ is a well-covered…