A new proof of the independence ratio of triangle-free cubic graphs

Staton proved that every triangle-free graph on n vertices with maximum degree three has an independent set of size at least 5n/14. A simpler proof was found by Jones. We give a yet simpler proof, and use it to design a linear-time algorithm to find such an independent set. Let G be a triangle-free graph on n vertices with maximum degree three. By Brooks’ theorem [1] G is 3-colorable; considering the largest color class, it follows that G has an independent set of size at least n/3. The coloring result is clearly best possible, but can we do better in terms of independent sets? Staton [6] proved that, in fact, the bound can be improved to 5n/14, and this is best possible because, as noticed by Fajtlowicz [2], the generalized Petersen graph P (7, 2) has fourteen vertices and no independent set of size six. Jones [5] found a simpler proof of Staton’s result. Griggs and Murphy [3] designed a linear-time algorithm to find an independent set in G of size at least 5(n − k)/14, where k is the number of components of G that are 3-regular. The objective of this paper is to give a yet simpler proof of Staton’s result, and to design a linear-time algorithm to find an independent set in G of size at least 5n/14. Graphs are finite and simple (that is, they have no loops or parallel edges). A block is either a 2-connected graph, or a complete graph on at most two vertices. A block of a graph G is a maximal subgraph of G that is a block. A pentagon is a cycle of length five. (Paths and cycles have no “repeated” vertices.) A block B is said to be difficult if it is isomorphic to a pentagon or L, where L is the graph obtained by subdividing both edges of a perfect matching of K4 twice; it has eight vertices, ten edges and independent sets of size three. A graph G is said to be difficult if every block of G is either difficult or is an edge between two difficult blocks. We define λ(G) to be the number of components of G that are difficult. Our main result reads as follows. Theorem. Every triangle-free graph G with maximum degree at most three has an independent set of size at least 1 7 (4 |V (G)| − |E(G)| − λ(G)) . Since every difficult component has at least two vertices of degree two, we deduce Staton’s theorem. Corollary. Every triangle-free graph on n vertices with maximum degree at most three has an independent set of size at least 5n/14. We offer the following conjecture, which would also imply the corollary. The fractional chromatic number of a graph G is the infimum of all a/b such that to every vertex of G one can assign a subset of {1, 2, . . . , a} of size b in such a way that adjacent vertices are assigned disjoint sets. It follows that the infimum is attained, because it is the optimum value of a certain linear program with rational data. The linear program is the linear programming relaxation of a certain integer program whose optimum is the chromatic number. It appears that the fractional chromatic number was first introduced in [4]. We conjecture the following. Conjecture. Every triangle-free graph with maximum degree at most three has fractional chromatic number at most 14/5. Proof of the theorem. To show the theorem holds, suppose for a contradiction that it does not, and let G be a counterexample with |V (G)| minimum. We proceed in a series of claims. (1) Let X ⊆ V (G) be nonempty, and let G be obtained from G\X by (possibly) adding edges so that no triangles or vertices of degree more than 3 are created. If every independent set I ′ in G can be extended to an independent set in G of size at least ∣