9 research outputs found
Problems on Matchings and Independent Sets of a Graph
Let be a finite simple graph. For , the difference of
, where is the neighborhood of and is called the critical difference of . is
called a critical set if equals the critical difference and ker is
the intersection of all critical sets. It is known that ker is an
independent (vertex) set of . diadem is the union of all critical
independent sets. An independent set is an inclusion minimal set with if no proper subset of has positive difference.
A graph is called K\"onig-Egerv\'ary if the sum of its independence
number () and matching number () equals . It is
known that bipartite graphs are K\"onig-Egerv\'ary.
In this paper, we study independent sets with positive difference for which
every proper subset has a smaller difference and prove a result conjectured by
Levit and Mandrescu in 2013. The conjecture states that for any graph, the
number of inclusion minimal sets with is at least the critical
difference of the graph. We also give a short proof of the inequality
kerdiadem (proved by Short in 2016).
A characterization of unicyclic non-K\"onig-Egerv\'ary graphs is also
presented and a conjecture which states that for such a graph , the critical
difference equals , is proved.
We also make an observation about ker using Edmonds-Gallai Structure
Theorem as a concluding remark.Comment: 18 pages, 2 figure
Some Extremal And Structural Problems In Graph Theory
This work considers three main topics. In Chapter 2, we deal with König-Egerváry graphs. We will give two new characterizations of König-Egerváry graphs as well as prove a related lower bound for the independence number of a graph. In Chapter 3, we study joint degree vectors (JDV). A problem arising from statistics is to determine the maximum number of non-zero elements of a JDV. We provide reasonable lower and upper bounds for this maximum number. Lastly, in Chapter 4 we study a problem in chemical graph theory. In particular, we characterize extremal cases for the number of maximal matchings in two linear polymers of chemical interest: the polyspiro chains and benzenoid chains. We also enumerate maximal matchings in several classes of these linear polymers and use the obtained results to determine the asymptotic behavior of these matchings