A Combinatorial Classic — Sparse Graphs with High Chromatic Number

Circular coloring is a constraints satisfaction problem where colors are assigned to nodes in a graph in such a way that every pair of connected nodes has two consecutive colors (the first color being consecutive to the last). We study circular coloring of random graphs using the cavity method. We identify two very interesting properties of this problem. For sufficiently many color and sufficiently low temperature there is a spontaneous breaking of the circular symmetry between colors and a phase transition forwards a ferromagnet-like phase. Our second main result concerns 5-circular coloring of random 3-regular graphs. While this case is found colorable, we conclude that the description via one-step replica symmetry breaking is not sufficient. We observe that simulated annealing is very efficient to find proper colorings for this case. The 5-circular coloring of 3-regular random graphs thus provides a first known example of a problem where the ground state energy is known to be exactly zero yet the space of solutions probably requires a full-step replica symmetry breaking treatment.

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“…"The Pentagon Problem" in [7]. The results of the cavity method confirm that random graphs of degree three are indeed 5-circular colorable.…”

Section: Introductionmentioning

confidence: 83%

“…so called girth at least l) then such a graph is 5-circular colorable [7]. This conjecture is inspired by the aim to generalize classical results known for coloring, for instance that every graph with maximum degree 3 is 3-colorable unless it contains a complete graph of 4 nodes [8].…”

Section: A Context In Mathematicsmentioning

confidence: 97%

Circular coloring of random graphs: statistical physics investigation

2016

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“…8 And the existence, for each p of graphs of maximal degree p, chromatic number larger that √ p/2 and of unbounded girth, see [35]. These graphs are not all in any class U q by Proposition 1.2(2).…”

Section: Tree-width and Clique-widthmentioning

confidence: 96%

On quasi-planar graphs: Clique-width and logical description

Courcelle

2020

Discrete Applied Mathematics

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Motivated by the construction of FPT graph algorithms parameterized by clique-width or tree-width, we study graph classes for which treewidth and clique-width are linearly related. This is the case for all graph classes of bounded expansion, but in view of concrete applications, we want to have "small" constants in the comparisons between these width parameters.We focus our attention on graphs that can be drawn in the plane with limited edge crossings, for an example, at most p crossings for each edge. These graphs are called p-planar. We consider a more general situation where the graph of edge crossings must belong to a fixed class of graphs D. For p-planar graphs, D is the class of graphs of degree at most p. We prove that tree-width and clique-width are linearly related for graphs drawable with a graph of crossings of bounded average degree.We prove that the class of 1-planar graphs, although conceptually close to that of planar graphs, is not characterized by a monadic second-order sentence. We identify two subclasses that are.

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“…Quite recently, Z. Li and B. Mohar [13] showed that the conjecture holds for digraphs of digirth at least 4. The second question is from P. Erdős and Neumann-Lara [6,15]:…”

Section: Proofmentioning

confidence: 99%