395 research outputs found
Planar graph coloring avoiding monochromatic subgraphs: trees and paths make things difficult
We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem
Minimal counterexamples and discharging method
Recently, the author found that there is a common mistake in some papers by
using minimal counterexample and discharging method. We first discuss how the
mistake is generated, and give a method to fix the mistake. As an illustration,
we consider total coloring of planar or toroidal graphs, and show that: if
is a planar or toroidal graph with maximum degree at most , where
, then the total chromatic number is at most .Comment: 8 pages. Preliminary version, comments are welcom
Every plane graph of maximum degree 8 has an edge-face 9-colouring
An edge-face colouring of a plane graph with edge set and face set is
a colouring of the elements of such that adjacent or incident
elements receive different colours. Borodin proved that every plane graph of
maximum degree can be edge-face coloured with colours.
Borodin's bound was recently extended to the case where . In this
paper, we extend it to the case .Comment: 29 pages, 1 figure; v2 corrects a contraction error in v1; to appear
in SIDM
Boundary chromatic polynomial
We consider proper colorings of planar graphs embedded in the annulus, such
that vertices on one rim can take Q_s colors, while all remaining vertices can
take Q colors. The corresponding chromatic polynomial is related to the
partition function of a boundary loop model. Using results for the latter, the
phase diagram of the coloring problem (with real Q and Q_s) is inferred, in the
limits of two-dimensional or quasi one-dimensional infinite graphs. We find in
particular that the special role played by Beraha numbers Q=4 cos^2(pi/n) for
the usual chromatic polynomial does not extend to the case Q different from
Q_s. The agreement with (scarce) existing numerical results is perfect; further
numerical checks are presented here.Comment: 20 pages, 7 figure
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