Speaker
Prof.
Donovan Hare
(University of British Columbia)
Description
The cycle space of a graph $G$ is the subspace of the edge space of $G$ over
the 2-element field that is
spanned by the cycles of $G$ (considered as edge-sets of $G$).
Bondy and Lov\'{a}sz (1981) showed that
the cycles through any set of $s-1$ vertices in an $s$-connected graph
generate its cycle space. But what if these cycles are restricted to be of odd length?
In this talk, we consider this question and others regarding the cycle bases
of nonbipartite graphs all of whose cycles are of odd length.
It will be proved that if the graph is 3-connected and nonbipartite,
then its cycle space can be generated by all odd cycles through a fix vertex.
An example of Toft (1975) is used to show that two fixed vertices cannot be specified,
contrasting the result of Bondy and Lov\'{a}sz when parity isn't a consideration.
Other related results regarding the number of odd cycles in nonbipartite graphs
will also be discussed.
Primary author
Prof.
Donovan Hare
(University of British Columbia)
