Abstract
The _independence ratio_ of a graph is the ratio of the size of its largest independent set to its number of vertices. Trivially, the independence ratio of a k-colorable graph is at least $1/k$ as each color class of a k-coloring is an independent set. However, better bounds can often be obtained for well-structured classes of graphs. In particular, Albertson, Bollobás and Tucker conjectured in 1976 that the independence ratio of every triangle-free subcubic planar graph is at least $3/8$. The conjecture was proven by Heckman and Thomas in 2006, and the ratio is best possible as there exists a cubic triangle-free planar graph with 24 vertices and the independence number equal to 9. The present article removes the planarity assumption. However, one needs to introduce an additional assumption since there are known to exist six 2-connected (non-planar) triangle-free subcubic graphs with the independence ratio less than $3/8$. Bajnok and Brinkmann conjectured that every 2-connected triangle-free subcubic graph has the independence ratio at least $3/8$ unless it is one of the six exceptional graphs. Fraughnaugh and Locke proposed a stronger conjecture: every triangle-free subcubic graph that does not contain one of the six exceptional graphs as a subgraph has independence ratio at least $3/8$. The authors prove these two conjectures, which implies in particular the result by Heckman and Thomas.
Highlights
All graphs in this paper are undirected, finite, and simple
The graph G is said to be subcubic if G has maximum degree at most 3, and cubic if G is 3-regular
Suppose that B is an 8−augmentation of a bad graph B that has a maximum independent set S avoiding all of its degree-2 vertices
Summary
All graphs in this paper are undirected, finite, and simple. A subset X of vertices of a graph G is independent if no two vertices in X are adjacent in G. One of the first results about independent sets in triangle-free subcubic graphs is the following theorem of Staton [19] from 1979. Locke [14] conjectured that there are only finitely many 3-connected triangle-free cubic n-vertex graphs. We would like to show that α(G) μ(G) holds under these hypotheses; the fact that vertices of degree less than 3 contribute more to the lower bound helps the induction go through This is not true because of bad graphs: An n-vertex bad graph G is connected, critical, has four degree-2 vertices, and yet α (G). Let G be a connected triangle-free subcubic n-vertex graph.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
