Finding large cycles in Hamiltonian graphs
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
On a simple randomized algorithm for finding a 2-factor in sparse graphs
Information Processing Letters
Approximating the maximum clique minor and some subgraph homeomorphism problems
Theoretical Computer Science
On a simple randomized algorithm for finding a 2-factor in sparse graphs
Information Processing Letters
Finding large cycles in Hamiltonian graphs
Discrete Applied Mathematics
Journal of Combinatorial Theory Series B
Approximating the longest cycle problem on graphs with bounded degree
COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
Hi-index | 0.00 |
We consider the problem of finding long paths and cycles in Hamiltonian graphs. The focus of our work is on sparse graphs, e.g., cubic graphs, that satisfy some property known to hold for Hamiltonian graphs, e.g., k-cyclability.We first consider the problem of finding long cycles in 3-connected cubic graphs whose edges have weights $w_i\geq 0$. We find cycles of weight at least ${(\sum w_i^a)}^{\frac{1}{a}}$ for $a=\log_2 3$. Based on this result, we develop an algorithm for finding a cycle of length at least $m^{(\log_3 2)/2}\approx m^{0.315}$ in 3-cyclable graphs with vertices of degree at most 3 and with m edges. As a corollary of this result, for arbitrary graphs with vertices of degree at most 3 that have a cycle of length l (or, more generally, a 3-cyclable minor with degrees at most 3 and with l edges), we find a cycle of length at least $l^{(\log_3 2)/2}$.We consider the graph property of 1-toughness that is common to Hamiltonian graphs and 3-connected cubic graphs, and we try to determine if 1-toughness implies the existence of long cycles. We show that 2-connectivity and 1-toughness, for constant degree graphs, may give cycles that are only of logarithmic length. However, we exhibit a class of 3-connected 1-tough graphs with degrees up to 6, where we can find cycles of length at least ${m}^{\log_3 2}/2$.