8 research outputs found
On Disjoint hypercubes in Fibonacci cubes
The {\em Fibonacci cube} of dimension , denoted as , is the
subgraph of -cube induced by vertices with no consecutive 1's. We
study the maximum number of disjoint subgraphs in isomorphic to
, and denote this number by . We prove several recursive results
for , in particular we prove that . We also prove a closed formula in which is given in
terms of Fibonacci numbers, and finally we give the generating function for the
sequence
Resonance graphs of plane bipartite graphs as daisy cubes
We characterize all plane bipartite graphs whose resonance graphs are daisy
cubes and therefore generalize related results on resonance graphs of benzenoid
graphs, catacondensed even ring systems, as well as 2-connected outerplane
bipartite graphs. Firstly, we prove that if is a plane elementary bipartite
graph other than , then the resonance graph is a daisy cube if and
only if the Fries number of equals the number of finite faces of , which
in turn is equivalent to being homeomorphically peripheral color
alternating. Next, we extend the above characterization from plane elementary
bipartite graphs to all plane bipartite graphs and show that the resonance
graph of a plane bipartite graph is a daisy cube if and only if is
weakly elementary bipartite and every elementary component of other than
is homeomorphically peripheral color alternating. Along the way, we prove
that a Cartesian product graph is a daisy cube if and only if all of its
nontrivial factors are daisy cubes