15 research outputs found
A note on circular chromatic number of graphs with large girth and similar problems
In this short note, we extend the result of Galluccio, Goddyn, and Hell,
which states that graphs of large girth excluding a minor are nearly bipartite.
We also prove a similar result for the oriented chromatic number, from which
follows in particular that graphs of large girth excluding a minor have
oriented chromatic number at most , and for the th chromatic number
, from which follows in particular that graphs of large girth
excluding a minor have
Chordal graphs with bounded tree-width
Given and , we prove that the number of labelled -connected chordal graphs with vertices and tree-width at most is asymptotically , as , for some constants depending on and . Additionally, we show that the number of -cliques () in a uniform random -connected chordal graph with tree-width at most is normally distributed as . The asymptotic enumeration of graphs of tree-width at most is wide open for . To the best of our knowledge, this is the first non-trivial class of graphs with bounded tree-width where the asymptotic counting problem is solved. Our starting point is the work of Wormald [Counting Labelled Chordal Graphs, \textit{Graphs and Combinatorics} (1985)], were an algorithm is developed to obtain the exact number of labelled chordal graphs on vertices.Peer ReviewedPostprint (author's final draft
Random graphs from a weighted minor-closed class
There has been much recent interest in random graphs sampled uniformly from
the n-vertex graphs in a suitable minor-closed class, such as the class of all
planar graphs. Here we use combinatorial and probabilistic methods to
investigate a more general model. We consider random graphs from a
`well-behaved' class of graphs: examples of such classes include all
minor-closed classes of graphs with 2-connected excluded minors (such as
forests, series-parallel graphs and planar graphs), the class of graphs
embeddable on any given surface, and the class of graphs with at most k
vertex-disjoint cycles. Also, we give weights to edges and components to
specify probabilities, so that our random graphs correspond to the random
cluster model, appropriately conditioned.
We find that earlier results extend naturally in both directions, to general
well-behaved classes of graphs, and to the weighted framework, for example
results concerning the probability of a random graph being connected; and we
also give results on the 2-core which are new even for the uniform (unweighted)
case.Comment: 46 page
Chordal graphs with bounded tree-width
Given and , we prove that the number of labelled
-connected chordal graphs with vertices and tree-width at most is
asymptotically , as , for some constants
depending on and . Additionally, we show that the number
of -cliques () in a uniform random -connected chordal
graph with tree-width at most is normally distributed as .
The asymptotic enumeration of graphs of tree-width at most is wide open
for . To the best of our knowledge, this is the first non-trivial
class of graphs with bounded tree-width where the asymptotic counting problem
is solved. Our starting point is the work of Wormald [Counting Labelled Chordal
Graphs, Graphs and Combinatorics (1985)], were an algorithm is developed to
obtain the exact number of labelled chordal graphs on vertices.Comment: 23 pages, 5 figure
A Separator Theorem for String Graphs and its Applications
A string graph is the intersection graph of a collection of continuous arcs in the plane. We show that any string graph with m edges can be separated into two parts of roughly equal size by the removal of vertices. This result is then used to deduce that every string graph with n vertices and no complete bipartite subgraph Kt,t has at most ctn edges, where ct is a constant depending only on t. Another application shows that locally tree-like string graphs are globally tree-like: for any Δ > 0, there is an integer g(Δ) such that every string graph with n vertices and girth at least g(Δ) has at most (1 + Δ)n edges. Furthermore, the number of such labelled graphs is at most (1 + Δ)nT(n), where T(n) = nnâ2 is the number of labelled trees on n vertice
On the purity of minor-closed classes of graphs
Given a graph H with at least one edge, let gapH(n) denote the maximum difference between the numbers of edges in two n-vertex edge-maximal graphs with no minor H. We show that for exactly four connected graphs H (with at least two vertices), the class of graphs with no minor H is pure, that is, gapH(n) = 0 for all n â„ 1; and for each connected graph H (with at least two vertices) we have the dichotomy that either gapH(n) = O(1) or gapH(n) = â(n). Further, if H is 2-connected and does not yield a pure class, then there is a constant c > 0 such that gapH(n) ⌠cn. We also give some partial results when H is not connected or when there are two or more excluded minors