2 research outputs found
Frank number and nowhere-zero flows on graphs
An edge of a graph is called deletable for some orientation if
the restriction of to is a strong orientation. Inspired by a problem
of Frank, in 2021 H\"orsch and Szigeti proposed a new parameter for
-edge-connected graphs, called the Frank number, which refines
-edge-connectivity. The Frank number is defined as the minimum number of
orientations of for which every edge of is deletable in at least one of
them. They showed that every -edge-connected graph has Frank number at most
and that in case these graphs are also -edge-colourable the parameter is
at most . Here we strengthen both results by showing that every
-edge-connected graph has Frank number at most and that every graph
which is -edge-connected and -edge-colourable has Frank number . The
latter also confirms a conjecture by Bar\'at and Bl\'azsik. Furthermore, we
prove two sufficient conditions for cubic graphs to have Frank number and
use them in an algorithm to computationally show that the Petersen graph is the
only cyclically -edge-connected cubic graph up to vertices having Frank
number greater than .Comment: 22 page
Generation and New Infinite Families of -hypohamiltonian Graphs
We present an algorithm which can generate all pairwise non-isomorphic
-hypohamiltonian graphs, i.e. non-hamiltonian graphs in which the removal
of any pair of adjacent vertices yields a hamiltonian graph, of a given order.
We introduce new bounding criteria specifically designed for
-hypohamiltonian graphs, allowing us to improve upon earlier computational
results. Specifically, we characterise the orders for which
-hypohamiltonian graphs exist and improve existing lower bounds on the
orders of the smallest planar and the smallest bipartite -hypohamiltonian
graphs. Furthermore, we describe a new operation for creating
-hypohamiltonian graphs that preserves planarity under certain conditions
and use it to prove the existence of a planar -hypohamiltonian graph of
order for every integer . Additionally, motivated by a theorem
of Thomassen on hypohamiltonian graphs, we show the existence
-hypohamiltonian graphs with large maximum degree and size.Comment: 21 page