Carlos S. Subi

Approximating the Longest Cycle Problem in Sparse Graphs

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

Finding long paths and cycles in sparse Hamiltonian graphs

w_i\geq 0

Maximum gap labelings of graphs

. We find cycles of weight at least

Edge-coloring almost bipartite multigraphs

{(\sum w_i^a)}^{\frac{1}{a}}

Disks on a Tree: Analysis of a Combinatorial Game

for

Nearly tight bounds on the number of Hamiltonian circuits of the hypercube and generalizations

a=\log_2 3

On hypercube labellings and antipodal monochromatic paths

. Based on this result, we develop an algorithm for finding a cycle of length at least

On Barnette's conjecture

m^{(\log_3 2)/2}\approx m^{0.315}

Packing Edge-Disjoint Triangles in Given Graphs.

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

Nearly Tight Bounds on the Number of Hamiltonian Circuits of the Hypercube and Generalizations (revised).

l^{(\log_3 2)/2}