Related papers: Perfect Pseudo-Matchings in cubic graphs

Two cycles are {\em adjacent} if they have an edge in common. Suppose that $G$ is a planar graph, for any two adjacent cycles $C_{1}$ and $C_{2}$, we have $|C_{1}| + |C_{2}| \geq 11$, in particular, when $|C_{1}| = 5$, $|C_{2}| \geq 7$. We…

Let alpha(G) denote the maximum size of an independent set of vertices and mu(G) be the cardinality of a maximum matching in a graph G. A matching saturating all the vertices is perfect. If alpha(G) + mu(G) equals the number of vertices of…

Let $G$ be a cubic graph which has a decomposition into a spanning tree $T$ and a $2$-regular subgraph $C$, i.e. $E(T) \cup E(C) = E(G)$ and $E(T) \cap E(C) = \emptyset$. We provide an answer to the following question: which lengths can the…

For many of the unsolved problems concerning cycles and matchings in graphs it is known that it is sufficient to prove them for \emph{snarks}, the class of nontrivial 3-regular graphs which cannot be 3-edge coloured. In the first part of…

This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are \emph{compatible} if they have the same vertex set and their union is also non-crossing. Our first result states that for any two perfect matchings…

A 2-matching of a graph $G$ is a spanning subgraph with maximum degree two. The size of a 2-matching $U$ is the number of edges in $U$ and this is at least $n-\k(U)$ where $n$ is the number of vertices of $G$ and $\k$ denotes the number of…

We study the 2-offer semirandom 3-uniform hypergraph model on $n$ vertices. At each step, we are presented with 2 uniformly random vertices. We choose any other vertex, thus creating a hyperedge of size 3. We show a strategy that constructs…

Let $X_{2k}$ be a set of $2k$ labeled points in convex position in the plane. We consider geometric non-intersecting straight-line perfect matchings of $X_{2k}$. Two such matchings, $M$ and $M'$, are disjoint compatible if they do not have…

Let $G$ be a graph. A total dominating set of $G$ is a set $S$ of vertices of $G$ such that every vertex is adjacent to at least one vertex in $S$. The total domatic number of a graph is the maximum number of total dominating sets which…

Let $k \in \mathbb{N}$ and $c \in \mathbb{Z}_k$, where $\mathbb{Z}_1=\mathbb{Z}$. A graph $G=(V(G),E(G))$ is said to be $c$-sum $k$-magic if there is a labeling $\ell:E(G) \rightarrow \mathbb{Z}_k \setminus \{0\}$ such that $\sum_{u \in…

A maximal planar graph is a graph which can be embedded in the plane such that every face of the graph is a triangle. The center of a graph is the subgraph induced by the vertices of minimum eccentricity. We introduce the notion of…

A connected graph $G$, of order two or more, is matching covered if each edge lies in some \pema. The tight cut decomposition of a matching covered graph $G$ yields a list of bricks and braces; as per a theorem of Lov{\'a}sz~\cite{lova87},…

A (possibly denerate) drawing of a graph $G$ in the plane is approximable by an embedding if it can be turned into an embedding by an arbitrarily small perturbation. We show that testing, whether a straight-line drawing of a planar graph…

We characterize hyperfinite bipartite graphings that admit measurable perfect matchings. In particular, we prove that every regular hyperfinite bipartite graphing admits a measurable perfect matching if it is one-ended or the degree is odd.…

The perfectly matchable subgraph polytope of a graph is a (0,1)-polytope associated with the vertex sets of matchings in the graph. In this paper, we study algebraic properties (compressedness, Gorensteinness) of the toric rings of…

We study the perfect matching lattice of a matching covered graph $G$, generated by the incidence vectors of its perfect matchings. Building on results of Lov\'asz and de Carvalho, Lucchesi, and Murty, we give a polynomial-time algorithm…

Let $G$ denote a distance-regular graph with diameter $D\ge 3$, valency $k$, and intersection numbers $a_i$, $b_i$, $c_i$. By a {\it pseudo cosine sequence} of $G$ we mean a sequence of real numbers $s_0, s_1, ..., s_D$ such that $s_0=1$…

The book embedding of a graph $G$ is to place the vertices of $G$ on the spine and draw the edges to the pages so that the edges in the same page do not cross with each other. A book embedding is matching if the vertices in the same page…

A graph is pseudo 2-factor isomorphic if all of its 2-factors have the same parity of number of cycles. Abreu et al. [J. Comb. Theory, Ser. B. 98 (2008) 432--442] conjectured that $K_{3,3}$, the Heawood graph and the Pappus graph are the…

Let $D$ be a digraph. Given a set of vertices $S \subseteq V(D)$, an $S$-path partition $\mathcal{P}$ of $D$ is a collection of paths of $D$ such that $\{V(P) \colon P \in \mathcal{P}\}$ is a partition of $V(D)$ and $|V(P) \cap S| = 1$ for…