2 research outputs found
Simple cubic graphs with no short traveling salesman tour
Let denote the length of a shortest travelling salesman tour in a
graph . We prove that for any , there exists a simple
-connected planar cubic graph such that , a simple -connected bipartite cubic graph
such that , and a simple
-connected cubic graph such that
Weak oddness as an approximation of oddness and resistance in cubic graphs
We introduce weak oddness , a new measure of
uncolourability of cubic graphs, defined as the least number of odd components
in an even factor. For every bridgeless cubic graph ,
, where denotes the
resistance of and denotes the oddness of , so this new
measure is an approximation of both oddness and resistance. We demonstrate that
there are graphs satisfying ,
and that the difference between any two of those three measures can be
arbitrarily large. The construction implies that if we replace a vertex of a
cubic graph with a triangle, then its oddness can decrease by an arbitrarily
large amount