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On facial unique-maximum (edge-)coloring
A facial unique-maximum coloring of a plane graph is a vertex coloring where
on each face the maximal color appears exactly once on the vertices of
. If the coloring is required to be proper, then the upper bound for
the minimal number of colors required for such a coloring is set to .
Fabrici and G\"oring [Fabrici and Goring 2016] even conjectured that colors
always suffice. Confirming the conjecture would hence give a considerable
strengthening of the Four Color Theorem. In this paper, we prove that the
conjecture holds for subcubic plane graphs, outerplane graphs and plane
quadrangulations. Additionally, we consider the facial edge-coloring analogue
of the aforementioned coloring and prove that every -connected plane graph
admits such a coloring with at most colors.Comment: 5 figure
Three-coloring triangle-free graphs on surfaces V. Coloring planar graphs with distant anomalies
We settle a problem of Havel by showing that there exists an absolute
constant d such that if G is a planar graph in which every two distinct
triangles are at distance at least d, then G is 3-colorable. In fact, we prove
a more general theorem. Let G be a planar graph, and let H be a set of
connected subgraphs of G, each of bounded size, such that every two distinct
members of H are at least a specified distance apart and all triangles of G are
contained in \bigcup{H}. We give a sufficient condition for the existence of a
3-coloring phi of G such that for every B\in H, the restriction of phi to B is
constrained in a specified way.Comment: 26 pages, no figures. Updated presentatio
Mixing Times of Markov Chains on Degree Constrained Orientations of Planar Graphs
We study Markov chains for -orientations of plane graphs, these are
orientations where the outdegree of each vertex is prescribed by the value of a
given function . The set of -orientations of a plane graph has
a natural distributive lattice structure. The moves of the up-down Markov chain
on this distributive lattice corresponds to reversals of directed facial cycles
in the -orientation. We have a positive and several negative results
regarding the mixing time of such Markov chains.
A 2-orientation of a plane quadrangulation is an orientation where every
inner vertex has outdegree 2. We show that there is a class of plane
quadrangulations such that the up-down Markov chain on the 2-orientations of
these quadrangulations is slowly mixing. On the other hand the chain is rapidly
mixing on 2-orientations of quadrangulations with maximum degree at most 4.
Regarding examples for slow mixing we also revisit the case of 3-orientations
of triangulations which has been studied before by Miracle et al.. Our examples
for slow mixing are simpler and have a smaller maximum degree, Finally we
present the first example of a function and a class of plane
triangulations of constant maximum degree such that the up-down Markov chain on
the -orientations of these graphs is slowly mixing
Difference of Facial Achromatic Numbers between Two Triangular Embeddings of a Graph
A facial -complete -coloring of a triangulation on a surface is a vertex -coloring such that every triple of -colors appears on the boundary of some face of . The facial -achromatic number of is the maximum integer such that has a facial -complete -coloring. This notion is an expansion of the complete coloring, that is, a proper vertex coloring of a graph such that every pair of colors appears on the ends of some edge.
For two triangulations and on a surface, may not be equal to even if is isomorphic to as graphs. Hence, it would be interesting to see how large the difference between and can be. We shall show that an upper bound for such difference in terms of the genus of the surface
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