2,987 research outputs found
The non-existence of certain regular graphs of girth 5
For certain positive integers k it is shown that there is no k-regular graph with girth 5 having k2 + 3 vertices. This provides a new lower bound for the number of vertices of girth 5 graphs with these valences.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/24359/1/0000628.pd
On graphs with cyclic defect or excess
The Moore bound constitutes both an upper bound on the order of a graph of
maximum degree and diameter and a lower bound on the order of a graph
of minimum degree and odd girth . Graphs missing or exceeding the
Moore bound by are called {\it graphs with defect or excess
}, respectively.
While {\it Moore graphs} (graphs with ) and graphs with defect or
excess 1 have been characterized almost completely, graphs with defect or
excess 2 represent a wide unexplored area.
Graphs with defect (excess) 2 satisfy the equation
(), where denotes the adjacency matrix of the graph in
question, its order, the matrix whose entries are all
1's, the adjacency matrix of a union of vertex-disjoint cycles, and
a polynomial with integer coefficients such that the matrix
gives the number of paths of length at most joining each pair
of vertices in the graph.
In particular, if is the adjacency matrix of a cycle of order we call
the corresponding graphs \emph{graphs with cyclic defect or excess}; these
graphs are the subject of our attention in this paper.
We prove the non-existence of infinitely many such graphs. As the highlight
of the paper we provide the asymptotic upper bound of
for the number of graphs of odd degree and cyclic defect or excess.
This bound is in fact quite generous, and as a way of illustration, we show the
non-existence of some families of graphs of odd degree and cyclic
defect or excess.
Actually, we conjecture that, apart from the M\"obius ladder on 8 vertices,
no non-trivial graph of any degree and cyclic defect or excess exists.Comment: 20 pages, 3 Postscript figure
Schnyder decompositions for regular plane graphs and application to drawing
Schnyder woods are decompositions of simple triangulations into three
edge-disjoint spanning trees crossing each other in a specific way. In this
article, we define a generalization of Schnyder woods to -angulations (plane
graphs with faces of degree ) for all . A \emph{Schnyder
decomposition} is a set of spanning forests crossing each other in a
specific way, and such that each internal edge is part of exactly of the
spanning forests. We show that a Schnyder decomposition exists if and only if
the girth of the -angulation is . As in the case of Schnyder woods
(), there are alternative formulations in terms of orientations
("fractional" orientations when ) and in terms of corner-labellings.
Moreover, the set of Schnyder decompositions on a fixed -angulation of girth
is a distributive lattice. We also show that the structures dual to
Schnyder decompositions (on -regular plane graphs of mincut rooted at a
vertex ) are decompositions into spanning trees rooted at such
that each edge not incident to is used in opposite directions by two
trees. Additionally, for even values of , we show that a subclass of
Schnyder decompositions, which are called even, enjoy additional properties
that yield a reduced formulation; in the case d=4, these correspond to
well-studied structures on simple quadrangulations (2-orientations and
partitions into 2 spanning trees). In the case d=4, the dual of even Schnyder
decompositions yields (planar) orthogonal and straight-line drawing algorithms.
For a 4-regular plane graph of mincut 4 with vertices plus a marked
vertex , the vertices of are placed on a grid according to a permutation pattern, and in the orthogonal drawing
each of the edges of has exactly one bend. Embedding
also the marked vertex is doable at the cost of two additional rows and
columns and 8 additional bends for the 4 edges incident to . We propose a
further compaction step for the drawing algorithm and show that the obtained
grid-size is strongly concentrated around for a uniformly
random instance with vertices
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