298,045 research outputs found
The equivariant topology of stable Kneser graphs
The stable Kneser graph , , , introduced by Schrijver
\cite{schrijver}, is a vertex critical graph with chromatic number , its
vertices are certain subsets of a set of cardinality . Bj\"orner and de
Longueville \cite{anders-mark} have shown that its box complex is homotopy
equivalent to a sphere, \Hom(K_2,SG_{n,k})\homot\Sphere^k. The dihedral group
acts canonically on , the group with 2 elements acts
on . We almost determine the -homotopy type of
\Hom(K_2,SG_{n,k}) and use this to prove the following results. The graphs
are homotopy test graphs, i.e. for every graph and such
that \Hom(SG_{2s,4},H) is -connected, the chromatic number
is at least . If and then
is not a homotopy test graph, i.e.\ there are a graph and an such
that \Hom(SG_{n,k}, G) is -connected and .Comment: 34 pp
Critical behavior in inhomogeneous random graphs
We study the critical behavior of inhomogeneous random graphs where edges are
present independently but with unequal edge occupation probabilities. The edge
probabilities are moderated by vertex weights, and are such that the degree of
vertex i is close in distribution to a Poisson random variable with parameter
w_i, where w_i denotes the weight of vertex i. We choose the weights such that
the weight of a uniformly chosen vertex converges in distribution to a limiting
random variable W, in which case the proportion of vertices with degree k is
close to the probability that a Poisson random variable with random parameter W
takes the value k. We pay special attention to the power-law case, in which
P(W\geq k) is proportional to k^{-(\tau-1)} for some power-law exponent \tau>3,
a property which is then inherited by the asymptotic degree distribution.
We show that the critical behavior depends sensitively on the properties of
the asymptotic degree distribution moderated by the asymptotic weight
distribution W. Indeed, when P(W\geq k) \leq ck^{-(\tau-1)} for all k\geq 1 and
some \tau>4 and c>0, the largest critical connected component in a graph of
size n is of order n^{2/3}, as on the Erd\H{o}s-R\'enyi random graph. When,
instead, P(W\geq k)=ck^{-(\tau-1)}(1+o(1)) for k large and some \tau\in (3,4)
and c>0, the largest critical connected component is of the much smaller order
n^{(\tau-2)/(\tau-1)}.Comment: 26 page
Large Non-Planar Graphs and an Application to Crossing-Critical Graphs
We prove that, for every positive integer k, there is an integer N such that
every 4-connected non-planar graph with at least N vertices has a minor
isomorphic to K_{4,k}, the graph obtained from a cycle of length 2k+1 by adding
an edge joining every pair of vertices at distance exactly k, or the graph
obtained from a cycle of length k by adding two vertices adjacent to each other
and to every vertex on the cycle. We also prove a version of this for
subdivisions rather than minors, and relax the connectivity to allow 3-cuts
with one side planar and of bounded size. We deduce that for every integer k
there are only finitely many 3-connected 2-crossing-critical graphs with no
subdivision isomorphic to the graph obtained from a cycle of length 2k by
joining all pairs of diagonally opposite vertices.Comment: To appear in Journal of Combinatorial Theory B. 20 pages. No figures.
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The minimum degree of minimal -factor-critical claw-free graphs*
A graph of order is said to be -factor-critical for integers
, if the removal of any vertices results in a graph with a
perfect matching. A -factor-critical graph is minimal if for every edge, the
deletion of it results in a graph that is not -factor-critical. In 1998, O.
Favaron and M. Shi conjectured that every minimal -factor-critical graph has
minimum degree . In this paper, we confirm the conjecture for minimal
-factor-critical claw-free graphs. Moreover, we show that every minimal
-factor-critical claw-free graph has at least
vertices of degree in the case of -connected, yielding further
evidence for S. Norine and R. Thomas' conjecture on the minimum degree of
minimal bricks when .Comment: 17 pages, 12 figure
2-crossing critical graphs with a V8 minor
The crossing number of a graph is the minimum number of pairwise crossings of edges among all planar drawings of the graph. A graph G is k-crossing critical if it has crossing number k and any proper subgraph of G has a crossing number less than k.
The set of 1-crossing critical graphs is is determined by Kuratowski’s Theorem to be {K5, K3,3}. Work has been done to approach the problem of classifying all 2-crossing critical graphs. The graph V2n is a cycle on 2n vertices with n intersecting chords. The only remaining graphs to find in the classification of 2-crossing critical graphs are those that are 3-connected with a V8 minor but no V10 minor.
This paper seeks to fill some of this gap by defining and completely describing a class of graphs called fully covered. In addition, we examine other ways in which graphs may be 2-crossing critical. This discussion classifies all known examples of 3-connected, 2-crossing critical graphs with a V8 minor but no V10 minor
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