Generalizing D-graphs

Graphs and Combinatorics, 2011

A super (d, )-regular graph on 2n vertices is a bipartite graph on the classes of vertices V 1 and V 2 , where |V 1 | = |V 2 | = n, in which the minimum degree and the maximum degree are between (d − )n and (d + )n, and for every

Ars Mathematica Contemporanea, 2021

A weak-odd edge coloring of a general digraph D is a (not necessarily proper) coloring of its edges such that for each vertex v ∈ V (D) at least one color c satisfies the following conditions: if d − D (v) > 0 then c appears an odd number of times on the incoming edges at v; and if d + D (v) > 0 then c appears an odd number of times on the outgoing edges at v. The minimum number of colors sufficient for a weak-odd edge coloring of D is the weak-odd chromatic index, denoted χ ′ wo (D). It is known that χ ′ wo (D) ≤ 3 for every digraph D, and that the bound is sharp. In this article we show that the weak-odd chromatic index can be determined in polynomial time. Restricting to edge colorings of D with at most two colors, the minimum number of vertices v ∈ V (D) for which no color c satisfies the above conditions is the defect of D, denoted def(D). Surprisingly, it turns out that the problem of determining the defect of digraphs is (polynomially) equivalent to the problem of finding the matching number of simple graphs. Moreover, we characterize the classes of associated digraphs and tournaments in terms of the weak-odd chromatic index and the defect.

Electronic Journal of Linear Algebra, 2012

This survey paper deals with upper and lower bounds on the number of k-matchings in regular graphs on N vertices. For the upper bounds we recall the upper matching conjecture which is known to hold for perfect matchings. For the lower bounds we first survey the known results for bipartite graphs, and their continuous versions as the van der Waerden and Tverberg permanent conjectures and its variants. We then discuss non-bipartite graphs. Little is known beyond the recent proof of the Lovász-Plummer conjecture on the exponential growth of perfect matchings in cubic bridgeless graphs. We discuss the problem of the minimum of haffnians on the convex set of matrices, whose extreme points are the adjacency matrices of subgraphs of the complete graph corresponding to perfect matchings. We also consider infinite regular graphs. The analog of k-matching is the p-monomer entropy, where p ∈ [0, 1] is the density of the number of matchings.

2011

Number of k -matchings in bipartite graphs and graphs as permanents and haffnians Upper bounds on permanents and haffnians: results and conjectures. Lower bounds on permanents and haffnians: results and conjectures. Shmuel Friedland Univ. Illinois at Chicago () Some open problems in matchings in graphs Stability, hyperbolicity, and zero localization of / 29 Matchings G = (V , E ) undirected graph with vertices V , edges E . matching in G: M ⊆ E no two edges in M share a common endpoint. e = (u, v) ∈ M is dimer v not covered by M is monomer. M called monomer-dimer cover of G. M is perfect matching ⇐⇒ no monomers. M is k-matching ⇐⇒ #M = k. Shmuel Friedland Univ. Illinois at Chicago () Some open problems in matchings in graphs Stability, hyperbolicity, and zero localization of / 29

2014

In this article we have derived the minimum order of an odd regular graph such that the graph has no matching. We have observed that how it is different from the case of even regular graphs. We have checked the consistency of the derived result with Petersen's theorem.

arXiv: Combinatorics, 2015

In 2009, Kong, Wang, and Lee introduced the problem of finding the edge-balanced index sets ($EBI$) of complete bipartite graphs $K_{m,n}$, where they examined the cases $n=1$, $2$, $3$, $4$, $5$ and the case $m=n$. Since then the problem of finding $EBI(K_{m,n})$, where $m \geq n$, has been completely resolved for the $m,n=$ odd, odd and odd, even cases. In this paper we find the edge-balanced index sets for complete bipartite graphs where both parts have even cardinality.

arXiv: Combinatorics, 2015

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\sum_{i=1}^n d(G,i) x^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. The $n$-barbell graph $Bar_n$ with $2n$ vertices, is formed by joining two copies of a complete graph $K_n$ by a single edge. We prove that for every $n\geq 2$, $Bar_n$ is not $\mathcal{D}$-unique, that is, there is another non-isomorphic graph with the same domination polynomial. More precisely, we show that for every $n$, the $\mathcal{D}$-equivalence class of barbell graph, $[Bar_n]$, contains many graphs, which one of them is the complement of book graph of order $n-1$, $B_{n-1}^c$. Also we present many families of graphs in $\mathcal{D}$-equivalence class of $K_{n_1}\cup K_{n_2}\cup \cdots\cup K_{n_k}$.

Discrete Mathematics, 1989

Hacettepe Journal of Mathematics and Statistics

Let $G=(V_G, E_G)$ be a simple and connected graph. A set $M\subseteq E_G$ is called a matching of $G$ if no two edges of $M$ are adjacent. The number of edges in $M$ is called its size. A matching $M$ is maximal if it cannot be extended to a larger matching in $G$. The smallest size of a maximal matching is called the saturation number of $G$. In this paper we are concerned with the saturation numbers of lexicographic product of graphs. We also address and solve an open problem about the size of maximum matchings in graphs with a given maximum degree $\Delta$.

The Electronic Journal of Combinatorics

For the set of graphs with a given degree sequence, consisting of any number of $2's$ and $1's$, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of $m$-matchings. We find the expected value of the number of $m$-matchings of $r$-regular bipartite graphs on $2n$ vertices with respect to the two standard measures. We state and discuss the conjectured upper and lower bounds for $m$-matchings in $r$-regular bipartite graphs on $2n$ vertices, and their asymptotic versions for infinite $r$-regular bipartite graphs. We prove these conjectures for $2$-regular bipartite graphs and for $m$-matchings with $m\le 4$.