
Defective and Clustered Colouring of Sparse Graphs
An (improper) graph colouring has "defect" d if each monochromatic subgr...
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The size Ramsey number of graphs with bounded treewidth
A graph G is Ramsey for a graph H if every 2colouring of the edges of G...
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EPGrepresentations with small gridsize
In an EPGrepresentation of a graph G each vertex is represented by a pa...
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Defective and Clustered Choosability of Sparse Graphs
An (improper) graph colouring has "defect" d if each monochromatic subgr...
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Glauber dynamics for colourings of chordal graphs and graphs of bounded treewidth
The Glauber dynamics on the colourings of a graph is a random process wh...
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TreewidthPliability and PTAS for MaxCSPs
We identify a sufficient condition, treewidthpliability, that gives a p...
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A local epsilon version of Reed's Conjecture
In 1998, Reed conjectured that every graph G satisfies χ(G) ≤1/2(Δ(G) + ...
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Clustered Variants of Hajós' Conjecture
Hajós conjectured that every graph containing no subdivision of the complete graph K_s+1 is properly scolorable. This result was disproved by Catlin. Indeed, the maximum chromatic number of such graphs is Ω(s^2/log s). In this paper we prove that O(s) colors are enough for a weakening of this conjecture that only requires every monochromatic component to have bounded size (socalled clustered coloring). Our approach in this paper leads to more results. Say that a graph is an almost (≤ 1)subdivision of a graph H if it can be obtained from H by subdividing edges, where at most one edge is subdivided more than once. We prove the following (where s ≥ 2): * Graphs of bounded treewidth and with no almost (≤ 1)subdivision of K_s+1 are schoosable with bounded clustering. * For every graph H, graphs with no Hminor and no almost (≤ 1)subdivision of K_s+1 are (s+1)colorable with bounded clustering. * For every graph H of maximum degree at most d, graphs with no Hsubdivision and no almost (≤ 1)subdivision of K_s+1 are max{s+3d5,2}colorable with bounded clustering. * For every graph H of maximum degree d, graphs with no K_s,t subgraph and no Hsubdivision are max{s+3d4,2}colorable with bounded clustering. * Graphs with no K_s+1subdivision are max{4s5,1}colorable with bounded clustering. The first result shows that the weakening of Hajós' conjecture is true for graphs of bounded treewidth in a stronger sense; the final result is the first O(s) bound on the clustered chromatic number of graphs with no K_s+1subdivision.
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