3,363 research outputs found
A new two-variable generalization of the chromatic polynomial
We present a two-variable polynomial, which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of a graph. This new polynomial satisfies both an edge decomposition formula and a vertex decomposition formula. We establish two general expressions for this new polynomial: one in terms of the broken circuit complex and one in terms of the lattice of forbidden colorings. We show that the new polynomial may be considered as a specialization of Stanley's chromatic symmetric function. We finally give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that it can be computed in polynomial time for trees and graphs of restricted pathwidth
The chromatic polynomial of fatgraphs and its categorification
Motivated by Khovanov homology and relations between the Jones polynomial and
graph polynomials, we construct a homology theory for embedded graphs from
which the chromatic polynomial can be recovered as the Euler characteristic.
For plane graphs, we show that our chromatic homology can be recovered from the
Khovanov homology of an associated link. We apply this connection with Khovanov
homology to show that the torsion-free part of our chromatic homology is
independent of the choice of planar embedding of a graph.
We extend our construction and categorify the Bollobas-Riordan polynomial (a
generalisation of the Tutte polynomial to embedded graphs). We prove that both
our chromatic homology and the Khovanov homology of an associated link can be
recovered from this categorification.Comment: A substantial revision. To appear in Advances in Mathematic
Families of Graphs With Chromatic Zeros Lying on Circles
We define an infinite set of families of graphs, which we call -wheels and
denote , that generalize the wheel () and biwheel ()
graphs. The chromatic polynomial for is calculated, and
remarkably simple properties of the chromatic zeros are found: (i) the real
zeros occur at for even and for odd;
and (ii) the complex zeros all lie, equally spaced, on the unit circle
in the complex plane. In the limit, the zeros
on this circle merge to form a boundary curve separating two regions where the
limiting function is analytic, viz., the exterior and
interior of the above circle. Connections with statistical mechanics are noted.Comment: 8 pages, Late
A bivariate chromatic polynomial for signed graphs
We study Dohmen--P\"onitz--Tittmann's bivariate chromatic polynomial
which counts all -colorings of a graph such
that adjacent vertices get different colors if they are . Our first
contribution is an extension of to signed graphs, for which we
obtain an inclusion--exclusion formula and several special evaluations giving
rise, e.g., to polynomials that encode balanced subgraphs. Our second goal is
to derive combinatorial reciprocity theorems for and its
signed-graph analogues, reminiscent of Stanley's reciprocity theorem linking
chromatic polynomials to acyclic orientations.Comment: 8 pages, 4 figure
Lattice Points in Orthotopes and a Huge Polynomial Tutte Invariant of Weighted Gain Graphs
A gain graph is a graph whose edges are orientably labelled from a group. A
weighted gain graph is a gain graph with vertex weights from an abelian
semigroup, where the gain group is lattice ordered and acts on the weight
semigroup. For weighted gain graphs we establish basic properties and we
present general dichromatic and forest-expansion polynomials that are Tutte
invariants (they satisfy Tutte's deletion-contraction and multiplicative
identities). Our dichromatic polynomial includes the classical graph one by
Tutte, Zaslavsky's two for gain graphs, Noble and Welsh's for graphs with
positive integer weights, and that of rooted integral gain graphs by Forge and
Zaslavsky. It is not a universal Tutte invariant of weighted gain graphs; that
remains to be found.
An evaluation of one example of our polynomial counts proper list colorations
of the gain graph from a color set with a gain-group action. When the gain
group is Z^d, the lists are order ideals in the integer lattice Z^d, and there
are specified upper bounds on the colors, then there is a formula for the
number of bounded proper colorations that is a piecewise polynomial function of
the upper bounds, of degree nd where n is the order of the graph.
This example leads to graph-theoretical formulas for the number of integer
lattice points in an orthotope but outside a finite number of affinographic
hyperplanes, and for the number of n x d integral matrices that lie between two
specified matrices but not in any of certain subspaces defined by simple row
equations.Comment: 32 pp. Submitted in 2007, extensive revisions in 2013 (!). V3: Added
references, clarified examples. 35 p
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