18 research outputs found
On the existence of -cages
In this paper, we work with simple and finite graphs. We study a
generalization of the \emph{Cage Problem}, which has been widely studied since
cages were introduced by Tutte \cite{T47} in 1947 and after Erd\" os and Sachs
\cite{ES63} proved their existence in 1963. An \emph{-graph} is an
-regular graph in which the shortest cycle has length equal to ; that is,
it is an -regular graph with girth . An \emph{-cage} is an
-graph with the smallest possible number of vertices among all
-graphs; the order of an -cage is denoted by . The Cage
Problem consists of finding -cages; it is well-known that -cages
have been determined only for very limited sets of parameter pairs .
There exists a simple lower bound for , given by Moore and denoted by
. The cages that attain this bound are called \emph{Moore cages}.Comment: 18 page
Antimagic Labelings of Weighted and Oriented Graphs
A graph is - if for any vertex weighting
and any list assignment with there exists an edge labeling
such that for all , labels of edges are pairwise
distinct, and the sum of the labels on edges incident to a vertex plus the
weight of that vertex is distinct from the sum at every other vertex. In this
paper we prove that every graph on vertices having no or
component is -weighted-list-antimagic.
An oriented graph is - if there exists an
injective edge labeling from into such that the
sum of the labels on edges incident to and oriented toward a vertex minus the
sum of the labels on edges incident to and oriented away from that vertex is
distinct from the difference of sums at every other vertex. We prove that every
graph on vertices with no component admits an orientation that is
-oriented-antimagic.Comment: 10 pages, 1 figur
Anti-van der Waerden Numbers of Graph Products with Trees
Given a graph , an exact -coloring of is a surjective function
. An arithmetic progression in of length with
common difference is a set of vertices such that
for . An arithmetic progression is rainbow
if all of the vertices are colored distinctly. The fewest number of colors that
guarantees a rainbow arithmetic progression of length three is called the
anti-van der Waerden number of and is denoted . It is known that
. Here we determine exact values for some trees and , determine for some trees
, and determine for some graphs and .Comment: 20 pages, 3 figure
Proof of a conjecture of Graham and Lov\'asz concerning unimodality of coefficients of the distance characteristic polynomial of a tree
We establish a conjecture of Graham and Lov\'asz that the (normalized)
coefficients of the distance characteristic polynomial of a tree are unimodal;
we also prove they are log-concave
On the Limiting Vacillating Tableaux for Integer Sequences
A fundamental identity in the representation theory of the partition algeba
is for , where
ranges over integer partitions of , is the number of
standard Young tableaux of shape , and is the number of
vacillating tableaux of shape and length . Using a combination of
RSK insertion and jeu de taquin, Halverson and Lewandowski constructed a
bijection that maps each integer sequence in to a pair
consisting of a standard Young tableau and a vacillating tableau. In this
paper, we show that for a given integer sequence , when is
sufficiently large, the vacillating tableaux determined by
become stable when ; the limit
is called the limiting vacillating tableau for . We give a
characterization of the set of limiting vacillating tableaux and presents
explicit formulas that enumerate those vacillating tableaux