Approximating the Circumference of 3-Connected Claw-free Graphs
Jackson and Wormald show that every 3-connected
K_1,d-free graph, on n vertices, contains a cycle of length at least 1/2 n^g(d) where g(d) = (log_2 6 + 2 log_2 (2d+1))^-1. For d = 3, g(d) ~ 0.122.
Improving this bound, we prove that if G is a 3-connected claw-free graph on at least 6 vertices, then there exists a cycle C in G such that |E(C)| is at least c n^g+5, where
g = log_3 2 and c > 1/7 is a constant.
To do this, we instead prove a stronger theorem that requires the cycle to contain two specified edges. We then use Tutte decomposition to partition the graph and then use the inductive
hypothesis of our theorem to find paths or cycles in the different parts of the decomposition.
Advisor:Duke, Richard; Yu, Xingxing; Vigoda, Eric; Thomas, Robin; Tetali, Prasad
School:Georgia Institute of Technology
School Location:USA - Georgia
Source Type:Master's Thesis
Date of Publication:08/25/2008