1,381 research outputs found
On the Enumeration of Certain Weighted Graphs
We enumerate weighted graphs with a certain upper bound condition. We also
compute the generating function of the numbers of these graphs, and prove that
it is a rational function. In particular, we show that if the given graph is a
bipartite graph, then its generating function is of the form
, where is the number of vertices of the graph
and is a polynomial of degree at most .Comment: 25 page
Combinatorics of bicubic maps with hard particles
We present a purely combinatorial solution of the problem of enumerating
planar bicubic maps with hard particles. This is done by use of a bijection
with a particular class of blossom trees with particles, obtained by an
appropriate cutting of the maps. Although these trees have no simple local
characterization, we prove that their enumeration may be performed upon
introducing a larger class of "admissible" trees with possibly doubly-occupied
edges and summing them with appropriate signed weights. The proof relies on an
extension of the cutting procedure allowing for the presence on the maps of
special non-sectile edges. The admissible trees are characterized by simple
local rules, allowing eventually for an exact enumeration of planar bicubic
maps with hard particles. We also discuss generalizations for maps with
particles subject to more general exclusion rules and show how to re-derive the
enumeration of quartic maps with Ising spins in the present framework of
admissible trees. We finally comment on a possible interpretation in terms of
branching processes.Comment: 41 pages, 19 figures, tex, lanlmac, hyperbasics, epsf. Introduction
and discussion/conclusion extended, minor corrections, references adde
Enumerating the edge-colourings and total colourings of a regular graph
In this paper, we are interested in computing the number of edge colourings and total colourings of a graph. We prove that the maximum number of -edge-colourings of a -regular graph on vertices is . Our proof is constructible and leads to a branching algorithm enumerating all the -edge-colourings of a -regular graph using a time and polynomial space. In particular, we obtain a algorithm on time and polynomial space to enumerate all the -edge colourings of a cubic graph, improving the running time of of the algorithm due to Golovach et al.~\cite{GKC10}. We also show that the number of -total-colourings of a connected cubic graph is at most . Again, our proof yields a branching algorithm to enumerate all the -total-colourings of a connected cubic graph
Structure and enumeration of (3+1)-free posets
A poset is (3+1)-free if it does not contain the disjoint union of chains of
length 3 and 1 as an induced subposet. These posets play a central role in the
(3+1)-free conjecture of Stanley and Stembridge. Lewis and Zhang have
enumerated (3+1)-free posets in the graded case by decomposing them into
bipartite graphs, but until now the general enumeration problem has remained
open. We give a finer decomposition into bipartite graphs which applies to all
(3+1)-free posets and obtain generating functions which count (3+1)-free posets
with labelled or unlabelled vertices. Using this decomposition, we obtain a
decomposition of the automorphism group and asymptotics for the number of
(3+1)-free posets.Comment: 28 pages, 5 figures. New version includes substantial changes to
clarify the construction of skeleta and the enumeration. An extended abstract
of this paper appears as arXiv:1212.535
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