A parallel minor is obtained from a graph by any sequence of edge contractions and parallel edge deletions. We prove that, for any positive integer k, every internally 4-connected graph of sufficiently high order contains a parallel minor isomorphic to a variation of K4,k with a complete graph on the vertices of degree k, the k-partition triple fan with a complete graph on the vertices of degree k, the k-spoke double wheel, the k-spoke double wheel with axle, the (2k+1)-rung Mobius zigzag ladder, the (2k)-rung zigzag ladder, or Kk. We also find the unavoidable parallel minors of 1-, 2-, and 3-connected graphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 313-326, 2009

/pdf/unavoidable-parallel-minors-of-4-connected-graphs-42kppd8wpr.pdf

Unavoidable parallel minors of 4-connected graphs

Facially-constrained colorings of plane graphs: A survey

Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic component has maximum degree at most $d$. A colouring has "clustering" $c$ if each monochromatic component has at most $c$ vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a given subgraph, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdi\`ere parameter, graphs with given circumference, graphs excluding a fixed graph as an immersion, graphs with given thickness, graphs with given stack- or queue-number, graphs excluding $K_t$ as a minor, graphs excluding $K_{s,t}$ as a minor, and graphs excluding an arbitrary graph $H$ as a minor. Several open problems are discussed.

Defective and Clustered Graph Colouring

For a graph G and an integer t we let mcc_t(G) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed family of graphs. We show that \mcc_2(G) = O(n^{2/3}) for any n-vertex graph G \in F. This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such F and every fixed t we show that mcc_t(G)=O(n^{2/(t+1)}). On the other hand we have examples of graphs G with no K_{t+3} minor and with mcc_t(G)=\Omega(n^{2/(2t-1)}). 
It is also interesting to consider graphs of bounded degrees. Haxell, Szabo, and Tardos proved \mcc_2(G) \leq 20000 for every graph G of maximum degree 5. We show that there are n-vertex 7-regular graphs G with \mcc_2(G)=\Omega(n), and more sharply, for every \epsilon>0 there exists c_\epsilon>0 and n-vertex graphs of maximum degree 7, average degree at most 6+\epsilon for all subgraphs, and with mcc_2(G)\ge c_\eps n. For 6-regular graphs it is known only that the maximum order of magnitude of \mcc_2 is between \sqrt n and n. 
We also offer a Ramsey-theoretic perspective of the quantity \mcc_t(G).

Graph coloring with no large monochromatic components

A coloring of a hypergraph's vertices is polychromatic if every hyperedge contains at least one vertex of each color; the polychromatic number is the maximum number of colors in such a coloring. Its dual, the cover-decomposition number, is the maximum number of disjoint hyperedge-covers. In geometric hypergraphs, there is extensive work on lower-bounding these numbers in terms of their trivial upper bounds (minimum hyperedge size and degree); our goal here is to broaden the study beyond geometric settings. We obtain algorithms yielding near-tight bounds for three families of hypergraphs: bounded hyperedge size, paths in trees, and bounded Vapnik--Chervonenkis (VC)-dimension. This reveals that discrepancy theory and iterated linear program relaxation are useful for cover-decomposition. Finally, we discuss the generalization of cover-decomposition to sensor cover.

Cover-Decomposition and Polychromatic Numbers

We show that the vertices of any plane graph in which every face is of size at least g can be colored by (3g Aý 5)=4 colors so that every color appears in every face. This is nearly tight, as there are plane graphs that admit no vertex coloring of this type with more than (3g+1)=4 colors. We further show that the problem of determining whether a plane graph admits a vertex coloring by 3 colors in which all colors appear in every face is NP-complete even for graphs in which all faces are of size 3 or 4 only. If all faces are of size 3 this can be decided in polynomial time.

/pdf/polychromatic-colorings-of-plane-graphs-zuilqz4v6f.pdf

Polychromatic colorings of plane graphs

/pdf/relaxed-two-coloring-of-cubic-graphs-3eiaje1g6l.pdf

Relaxed two-coloring of cubic graphs

We show that any graph of maximum degree at most $3$ has a two-coloring, such that one color-class is an independent set while the other color induces monochromatic components of order at most $189$. On the other hand for any constant $C$ we exhibit a $4$-regular graph, such that the deletion of any independent set leaves at least one component of order greater than $C$. Similar results are obtained for coloring graphs of given maximum degree with $k+ \ell$ colors such that $k$ parts form an independent set and $\ell$ parts span components of order bounded by a constant. A lot of interesting questions remain open.

Relaxed Two-Coloring of Cubic Graphs

A coloring is proper if each color class induces connected components of order one (where the order of a graph is its number of vertices). Here we study relaxations of proper two-colorings, such that the order of the induced monochromatic components in one (or both) of the color classes is bounded by a constant. In a (C1, C2)-relaxed coloring of a graph G every monochromatic component induced by vertices of the first (second) color is of order at most C1 (C2, resp.). We are mostly concerned with (1,C)-relaxed colorings, in other words when/how is it possible to break up a graph into small components with the removal of an independent set.

We prove that every graph of maximum degree at most three can be (1,22)-relaxed colored and we give a quasilinear algorithm which constructs such a coloring. We also show that a similar statement cannot be true for graphs of maximum degree at most 4 in a very strong sense: we construct 4-regular graphs such that the removal of any independent set leaves a connected component whose order is linear in the number of vertices.

Furthermore we investigate the complexity of the decision problem (Δ, C)- AsymRelCol: Given a graph of maximum degree at most Δ, is there a (1,C)- relaxed coloring of G? We find a remarkable hardness jump in the behavior of this problem. We note that there is not even an obvious monotonicity in the hardness of the problem as C grows, i.e. the hardness for component order C+1 does not imply directly the hardness for C. In fact for C=1 the problem is obviously polynomial-time decidable, while it is shown that it is NP-hard for C=2 and Δ≥3.

For arbitrary Δ≥2 we still establish the monotonicity of hardness of (Δ, C)- AsymRelCol on the interval 2≤C≤f∞ in the following strong sense. There exists a critical component order f(Δ){∞} such that the problem of deciding (1,C)-relaxed colorability of graphs of maximum degree at most Δ is NP-complete for every 2≤C<f(Δ), while deciding (1,f(Δ))-colorability is trivial: every graph of maximum degree Δ is (1,f(Δ))-colorable. For Δ=3 the existence of this threshold is shown despite the fact that we do not know its precise value, only 6≤f(3)≤22. For any Δ≤4, (Δ, C)-AsymRelCol is NP-complete for arbitrary C≥2, so f(Δ)=∞.

We also study the symmetric version of the relaxed coloring problem, and make the first steps towards establishing a similar hardness jump.

/pdf/deciding-relaxed-two-colorability-a-hardness-jump-3byq0eqzf0.pdf

Deciding relaxed two-colorability: a hardness jump

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Papers

Polychromatic colorings of plane graphs

Relaxed two-coloring of cubic graphs

Relaxed Two-Coloring of Cubic Graphs

Deciding relaxed two-colorability: a hardness jump