Independence, odd girth, and average degree

Abstract

We prove several best-possible lower bounds in terms of the order and the average degree for the independence number of graphs which are connected and/or satisfy some odd girth condition. Our main result is the extension of a lower bound for the independence number of triangle-free graphs of maximum degree at most 3 due to Heckman and Thomas [A New Proof of the Independence Ratio of Triangle-Free Cubic Graphs, Discrete Math. 233 (2001), 233-237] to arbitrary triangle-free graphs. For connected triangle-free graphs of order n and size m, our result implies the existence of an independent set of order at least (4n − m − 1)/7.

Uniquely restricted matchings in subcubic graphs

“…The search for bounds in graphs of bounded degree and large girth is a common task for various hard graph invariants, cf. [1,2,6,7,10,11]. In [4] we proved that…”

Some bounds on the size of Maximum $G$-free sets in graph

“…This problem was solved by Erdös, and after that by Moon and Moser in [13]. There are very few works about counting the number of maximum independent sets, see [8][9][10]12] and [10]. Finding a maximum independent set in a graph is a NP-hard problem Another version of the independence number is the forest number of a graph.…”

Uniquely restricted matchings in subcubic graphs without short cycles

“…The search for bounds in graphs of bounded degree and large girth is a common task for various hard graph invariants, cf. [1,2,6,7,10,11]. In [4] we proved that for every connected graph of order , maximum degree at most , and girth at least 5, provided that .…”