22,223 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
On the Existence of General Factors in Regular Graphs
Let be a graph, and a set function
associated with . A spanning subgraph of is called an -factor if
the degree of any vertex in belongs to the set . This paper
contains two results on the existence of -factors in regular graphs. First,
we construct an -regular graph without some given -factor. In
particular, this gives a negative answer to a problem recently posed by Akbari
and Kano. Second, by using Lov\'asz's characterization theorem on the existence
of -factors, we find a sharp condition for the existence of general
-factors in -graphs, in terms of the maximum and minimum of .
The result reduces to Thomassen's theorem for the case that consists of
the same two consecutive integers for all vertices , and to Tutte's theorem
if the graph is regular in addition.Comment: 10 page
Vertex-Coloring Edge-Weighting of Bipartite Graphs with Two Edge Weights
Let be a graph and be a subset of . A vertex-coloring
-edge-weighting of is an assignment of weight by the
elements of to each edge of so that adjacent vertices have
different sums of incident edges weights.
It was proved that every 3-connected bipartite graph admits a vertex-coloring
-edge-weighting (Lu, Yu and Zhang, (2011) \cite{LYZ}). In this paper,
we show that the following result: if a 3-edge-connected bipartite graph
with minimum degree contains a vertex such that
and is connected, then admits a vertex-coloring
-edge-weighting for . In
particular, we show that every 2-connected and 3-edge-connected bipartite graph
admits a vertex-coloring -edge-weighting for . The bound is sharp, since there exists a family of
infinite bipartite graphs which are 2-connected and do not admit
vertex-coloring -edge-weightings or vertex-coloring
-edge-weightings.Comment: In this paper, we show that every 2-connected and 3-edge-connected
bipartite graph admits a vertex-coloring S-edge-weighting for S\in
{{0,1},{1,2}
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