2,708 research outputs found
Chromatic roots and minor-closed families of graphs
Given a minor-closed class of graphs , what is the infimum of
the non-trivial roots of the chromatic polynomial of ? When
is the class of all graphs, the answer is known to be . We
answer this question exactly for three minor-closed classes of graphs.
Furthermore, we conjecture precisely when the value is larger than .Comment: 18 pages, 5 figure
Chromatic roots are dense in the whole complex plane
I show that the zeros of the chromatic polynomials P-G(q) for the generalized theta graphs Theta((s.p)) are taken together, dense in the whole complex plane with the possible exception of the disc \q - l\ < l. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) Z(G)(q,upsilon) outside the disc \q + upsilon\ < \upsilon\. An immediate corollary is that the chromatic roots of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha-Kahane-Weiss theorem oil the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof
Grad and classes with bounded expansion I. decompositions
We introduce classes of graphs with bounded expansion as a generalization of
both proper minor closed classes and degree bounded classes. Such classes are
based on a new invariant, the greatest reduced average density (grad) of G with
rank r, grad r(G). For these classes we prove the existence of several
partition results such as the existence of low tree-width and low tree-depth
colorings. This generalizes and simplifies several earlier results (obtained
for minor closed classes)
Chromatic roots are dense in the whole complex plane
I show that the zeros of the chromatic polynomials P_G(q) for the generalized
theta graphs \Theta^{(s,p)} are, taken together, dense in the whole complex
plane with the possible exception of the disc |q-1| < 1. The same holds for
their dichromatic polynomials (alias Tutte polynomials, alias Potts-model
partition functions) Z_G(q,v) outside the disc |q+v| < |v|. An immediate
corollary is that the chromatic zeros of not-necessarily-planar graphs are
dense in the whole complex plane. The main technical tool in the proof of these
results is the Beraha-Kahane-Weiss theorem on the limit sets of zeros for
certain sequences of analytic functions, for which I give a new and simpler
proof.Comment: LaTeX2e, 53 pages. Version 2 includes a new Appendix B. Version 3
adds a new Theorem 1.4 and a new Section 5, and makes several small
improvements. To appear in Combinatorics, Probability & Computin
Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids
The chromatic polynomial P_G(q) of a loopless graph G is known to be nonzero
(with explicitly known sign) on the intervals (-\infty,0), (0,1) and (1,32/27].
Analogous theorems hold for the flow polynomial of bridgeless graphs and for
the characteristic polynomial of loopless matroids. Here we exhibit all these
results as special cases of more general theorems on real zero-free regions of
the multivariate Tutte polynomial Z_G(q,v). The proofs are quite simple, and
employ deletion-contraction together with parallel and series reduction. In
particular, they shed light on the origin of the curious number 32/27.Comment: LaTeX2e, 49 pages, includes 5 Postscript figure
Is the five-flow conjecture almost false?
The number of nowhere zero Z_Q flows on a graph G can be shown to be a
polynomial in Q, defining the flow polynomial \Phi_G(Q). According to Tutte's
five-flow conjecture, \Phi_G(5) > 0 for any bridgeless G.A conjecture by Welsh
that \Phi_G(Q) has no real roots for Q \in (4,\infty) was recently disproved by
Haggard, Pearce and Royle. These authors conjectured the absence of roots for Q
\in [5,\infty). We study the real roots of \Phi_G(Q) for a family of non-planar
cubic graphs known as generalised Petersen graphs G(m,k). We show that the
modified conjecture on real flow roots is also false, by exhibiting infinitely
many real flow roots Q>5 within the class G(nk,k). In particular, we compute
explicitly the flow polynomial of G(119,7), showing that it has real roots at
Q\approx 5.0000197675 and Q\approx 5.1653424423. We moreover prove that the
graph families G(6n,6) and G(7n,7) possess real flow roots that accumulate at
Q=5 as n\to\infty (in the latter case from above and below); and that
Q_c(7)\approx 5.2352605291 is an accumulation point of real zeros of the flow
polynomials for G(7n,7) as n\to\infty.Comment: 44 pages (LaTeX2e). Includes tex file, three sty files, and a
mathematica script polyG119_7.m. Many improvements from version 3, in
particular Sections 3 and 4 have been mostly re-writen, and Sections 7 and 8
have been eliminated. (This material can now be found in arXiv:1303.5210.)
Final version published in J. Combin. Theory
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