16,907 research outputs found
On realization graphs of degree sequences
Given the degree sequence of a graph, the realization graph of is the
graph having as its vertices the labeled realizations of , with two vertices
adjacent if one realization may be obtained from the other via an
edge-switching operation. We describe a connection between Cartesian products
in realization graphs and the canonical decomposition of degree sequences
described by R.I. Tyshkevich and others. As applications, we characterize the
degree sequences whose realization graphs are triangle-free graphs or
hypercubes.Comment: 10 pages, 5 figure
Some results on the structure of multipoles in the study of snarks
Multipoles are the pieces we obtain by cutting some edges of a cubic graph.
As a result of the cut, a multipole has dangling edges with one free end,
which we call semiedges. Then, every 3-edge-coloring of a multipole induces a
coloring or state of its semiedges, which satisfies the Parity Lemma.
Multipoles have been extensively used in the study of snarks, that is, cubic
graphs which are not 3-edge-colorable. Some results on the states and structure
of the so-called color complete and color closed multipoles are presented. In
particular, we give lower and upper linear bounds on the minimum order of a
color complete multipole, and compute its exact number of states. Given two
multipoles and with the same number of semiedges, we say that
is reducible to if the state set of is a non-empty subset of the
state set of and has less vertices than . The function
is defined as the maximum number of vertices of an irreducible multipole with
semiedges. The exact values of are only known for . We prove
that tree and cycle multipoles are irreducible and, as a byproduct, that
has a linear lower bound
- …