Size and independence in triangle-free graphs with maximum degree three
Journal of Graph Theory
A new proof of the independence ratio of triangle-free cubic graphs
Discrete Mathematics
Node-and edge-deletion NP-complete problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Triangle-free subcubic graphs with minimum bipartite density
Journal of Combinatorial Theory Series B
Bipartite density of triangle-free subcubic graphs
Discrete Applied Mathematics
The Fractional Chromatic Number of Graphs of Maximum Degree at Most Three
SIAM Journal on Discrete Mathematics
The Fractional Chromatic Number of Graphs of Maximum Degree at Most Three
SIAM Journal on Discrete Mathematics
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Suppose G is a graph with n vertices and m edges. Let n^' be the maximum number of vertices in an induced bipartite subgraph of G and let m^' be the maximum number of edges in a spanning bipartite subgraph of G. Then b(G)=m^'/m is called the bipartite density of G, and b^*(G)=n^'/n is called the bipartite ratio of G. This paper proves that every 2-connected triangle-free subcubic graph, apart from seven exceptions, has b(G)=17/21. Every 2-connected triangle-free subcubic graph other than the Petersen graph and the dodecahedron has b^*(G)=5/7. The bounds that b^*(G)=5/7 and b(G)=17/21 are tight in the sense that there are infinitely many 2-connected triangle-free cubic graphs G for which b(G)=17/21 and b^*(G)=5/7. On the other hand, if G is not cubic (i.e., G have vertices of degree at most 2), then the strict inequalities b^*(G)5/7 and b(G)17/21 hold, with a few exceptions. Nevertheless, the bounds are still sharp in the sense that for any @e0, there are infinitely many 2-connected subcubic graphs G with minimum degree 2 such that b^*(G)=17/21 is a common improvement of an earlier result of Bondy and Locke and a recent result of Xu and Yu: Bondy and Locke proved that every triangle-free cubic graph other than the Petersen graph and the dodecahedron has b(G)4/5. Xu and Yu confirmed a conjecture of Bondy and Locke and proved that every 2-connected triangle free subcubic graph with minimum degree 2 apart from five exceptions has b(G)4/5. The bound b^*(G)=5/7 is a strengthening of a well-known result (first proved by Fajtlowicz and by Staton, and with a few new proofs found later) which says that any triangle-free subcubic graph G has independence ratio at least 5/14. The proofs imply a linear time algorithm that finds an induced bipartite subgraph H of G with |V(H)|/|V(G)|=5/7 and a spanning bipartite subgraph H^' of G with |E(H^')|/|E(G)|=17/21.