1,042 research outputs found
A construction for a counterexample to the pseudo 2-factor isomorphic graph conjecture
A graph admiting a -factor is \textit{pseudo -factor isomorphic} if
the parity of the number of cycles in all its -factors is the same. In [M.
Abreu, A.A. Diwan, B. Jackson, D. Labbate and J. Sheehan. Pseudo -factor
isomorphic regular bipartite graphs. Journal of Combinatorial Theory, Series B,
98(2) (2008), 432-444.] some of the authors of this note gave a partial
characterisation of pseudo -factor isomorphic bipartite cubic graphs and
conjectured that , the Heawood graph and the Pappus graph are the only
essentially -edge-connected ones. In [J. Goedgebeur. A counterexample to the
pseudo -factor isomorphic graph conjecture. Discr. Applied Math., 193
(2015), 57-60.] Jan Goedgebeur computationally found a graph on
vertices which is pseudo -factor isomorphic cubic and bipartite,
essentially -edge-connected and cyclically -edge-connected, thus refuting
the above conjecture. In this note, we describe how such a graph can be
constructed from the Heawood graph and the generalised Petersen graph
, which are the Levi graphs of the Fano configuration and the
M\"obius-Kantor configuration, respectively. Such a description of
allows us to understand its automorphism group, which has order
, using both a geometrical and a graph theoretical approach
simultaneously. Moreover we illustrate the uniqueness of this graph
Irreducible pseudo 2-factor isomorphic cubic bipartite graphs
A bipartite graph is {\em pseudo 2--factor isomorphic} if all its 2--factors
have the same parity of number of circuits. In \cite{ADJLS} we proved that the
only essentially 4--edge-connected pseudo 2--factor isomorphic cubic bipartite
graph of girth 4 is , and conjectured \cite[Conjecture 3.6]{ADJLS}
that the only essentially 4--edge-connected cubic bipartite graphs are
, the Heawood graph and the Pappus graph.
There exists a characterization of symmetric configurations %{\bf
decide notation and how to use it in the rest of the paper} due to Martinetti
(1886) in which all symmetric configurations can be obtained from an
infinite set of so called {\em irreducible} configurations \cite{VM}. The list
of irreducible configurations has been completed by Boben \cite{B} in terms of
their {\em irreducible Levi graphs}.
In this paper we characterize irreducible pseudo 2--factor isomorphic cubic
bipartite graphs proving that the only pseudo 2--factor isomorphic irreducible
Levi graphs are the Heawood and Pappus graphs. Moreover, the obtained
characterization allows us to partially prove the above Conjecture
Embedability between right-angled Artin groups
In this article we study the right-angled Artin subgroups of a given
right-angled Artin group. Starting with a graph \gam, we produce a new graph
through a purely combinatorial procedure, and call it the extension graph
\gam^e of \gam. We produce a second graph \gam^e_k, the clique graph of
\gam^e, by adding extra vertices for each complete subgraph of \gam^e. We
prove that each finite induced subgraph of \gam^e gives rise to an
inclusion A(\Lambda)\to A(\gam). Conversely, we show that if there is an
inclusion A(\Lambda)\to A(\gam) then is an induced subgraph of
\gam^e_k. These results have a number of corollaries. Let denote the
path on four vertices and let denote the cycle of length . We prove
that embeds in A(\gam) if and only if is an induced subgraph
of \gam. We prove that if is any finite forest then embeds in
. We recover the first author's result on co--contraction of graphs and
prove that if \gam has no triangles and A(\gam) contains a copy of
for some , then \gam contains a copy of for some . We also recover Kambites' Theorem, which asserts that if embeds in
A(\gam) then \gam contains an induced square. Finally, we determine
precisely when there is an inclusion and show that there is
no "universal" two--dimensional right-angled Artin group.Comment: 35 pages. Added an appendix and a proof that the extension graph is
quasi-isometric to a tre
A counterexample to the pseudo 2-factor isomorphic graph conjecture
A graph is pseudo 2-factor isomorphic if the parity of the number of
cycles in a 2-factor is the same for all 2-factors of . Abreu et al.
conjectured that , the Heawood graph and the Pappus graph are the only
essentially 4-edge-connected pseudo 2-factor isomorphic cubic bipartite graphs
(Abreu et al., Journal of Combinatorial Theory, Series B, 2008, Conjecture
3.6).
Using a computer search we show that this conjecture is false by constructing
a counterexample with 30 vertices. We also show that this is the only
counterexample up to at least 40 vertices.
A graph is 2-factor hamiltonian if all 2-factors of are hamiltonian
cycles. Funk et al. conjectured that every 2-factor hamiltonian cubic bipartite
graph can be obtained from and the Heawood graph by applying repeated
star products (Funk et al., Journal of Combinatorial Theory, Series B, 2003,
Conjecture 3.2). We verify that this conjecture holds up to at least 40
vertices.Comment: 8 pages, added some extra information in Discrete Applied Mathematics
(2015
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Taut distance-regular graphs and the subconstituent algebra
We consider a bipartite distance-regular graph with diameter at least
4 and valency at least 3. We obtain upper and lower bounds for the local
eigenvalues of in terms of the intersection numbers of and the
eigenvalues of . Fix a vertex of and let denote the corresponding
subconstituent algebra. We give a detailed description of those thin
irreducible -modules that have endpoint 2 and dimension . In an earlier
paper the first author defined what it means for to be taut. We obtain
three characterizations of the taut condition, each of which involves the local
eigenvalues or the thin irreducible -modules mentioned above.Comment: 29 page
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