2 research outputs found
Zero Forcing in Claw-Free Cubic Graphs
The zero forcing number of a simple graph, written , is a NP-hard graph
invariant which is the result of the zero forcing color change rule. This graph
invariant has been heavily studied by linear algebraists, physicists, and graph
theorist. It's broad applicability and interesting combinatorial properties
have attracted the attention of many researchers. Of particular interest, is
that of bounding the zero forcing number from above. In this paper we show a
surprising relation between the zero forcing number of a graph and the
independence number of a graph, denoted . Our main theorem states
that if is a connected, cubic, claw-free graph, then . This improves on best known upper bounds for , as well as
known lower bounds on . As a consequence of this result, if is a connected, cubic, claw-free graph with order , then . Additionally, under the hypothesis of our main theorem, we
further show , where denotes the matching
number of
Parity Binomial Edge Ideals with Pure Resolutions
We provide a characterisation of all graphs whose parity binomial edge ideals
have pure resolutions. In particular, we show that the minimal free resolution
of a parity binomial edge ideal is pure if and only if the corresponding graph
is a complete bipartite graph, or a disjoint union of paths and odd cycles.Comment: 13 pages, 6 figure