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Restricted colorings of graphs
 in Surveys in Combinatorics 1993, London Math. Soc. Lecture Notes Series 187
, 1993
"... The problem of properly coloring the vertices (or edges) of a graph using for each vertex (or edge) a color from a prescribed list of permissible colors, received a considerable amount of attention. Here we describe the techniques applied in the study of this subject, which combine combinatorial, al ..."
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Cited by 76 (15 self)
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The problem of properly coloring the vertices (or edges) of a graph using for each vertex (or edge) a color from a prescribed list of permissible colors, received a considerable amount of attention. Here we describe the techniques applied in the study of this subject, which combine combinatorial, algebraic and probabilistic methods, and discuss several intriguing conjectures and open problems. This is mainly a survey of recent and less recent results in the area, but it contains several new results as well.
Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring
 In 46th Annual IEEE Symposium on Foundations of Computer Science
, 2005
"... At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topolog ..."
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Cited by 47 (12 self)
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At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a cliquesum of pieces almostembeddable into boundedgenus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2approximation to graph coloring, constantfactor approximations to treewidth and the largest grid minor, combinatorial polylogarithmicapproximation to halfintegral multicommodity flow, subexponential fixedparameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor. 1.
Hadwiger’s conjecture for K6free graphs
 COMBINATORICA
, 1993
"... In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t + 1 vertices is tcolourable. When t ≤ 3 this is easy, and when t = 4, Wagner’s theorem of 1937 shows the conjecture to be equivalent to the fourcolour conjecture (the 4CC). However, when t ..."
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Cited by 36 (2 self)
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In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t + 1 vertices is tcolourable. When t ≤ 3 this is easy, and when t = 4, Wagner’s theorem of 1937 shows the conjecture to be equivalent to the fourcolour conjecture (the 4CC). However, when t ≥ 5 it has remained open. Here we show that when t = 5 it is also equivalent to the 4CC. More precisely, we show (without assuming the 4CC) that every minimal counterexample to Hadwiger’s conjecture when t = 5 is “apex”, that is, it consists of a planar graph with one additional vertex. Consequently, the 4CC implies Hadwiger’s conjecture when t = 5, because it implies that apex graphs are 5colourable.
Recent Excluded Minor Theorems for Graphs
 IN SURVEYS IN COMBINATORICS, 1999 267 201222. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 (2001), #R34 8
, 1999
"... A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We disc ..."
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Cited by 9 (0 self)
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A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem of Robertson and Seymour, linkless embeddings of graphs in 3space, Hadwiger’s conjecture on tcolorability of graphs with no Kt+1 minor, Tutte’s edge 3coloring conjecture on edge 3colorability of 2connected cubic graphs with no Petersen minor, and Pfaffian orientations of bipartite graphs. The latter are related to the even directed circuit problem, a problem of Pólya about permanents, the 2colorability of hypergraphs, and signnonsingular matrices.
Independent sets in graphs with an excluded clique minor
 Discrete Math. Theor. Comput. Sci
"... Let G be a graph with n vertices, with independence number α, and with no Kt+1minor for some t ≥ 5. It is proved that (2α − 1)(2t − 5) ≥ 2n − 5. This improves upon the previous best bound whenever n ≥ 2 5 t2. ..."
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Cited by 6 (0 self)
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Let G be a graph with n vertices, with independence number α, and with no Kt+1minor for some t ≥ 5. It is proved that (2α − 1)(2t − 5) ≥ 2n − 5. This improves upon the previous best bound whenever n ≥ 2 5 t2.
Graph Color Extensions: When Hadwiger's Conjecture and Embeddings Help
 Electronic J. Comb
, 2002
"... Suppose G is rcolorable and P V (G) is such that the components of G[P ] are far apart. We show that any (r + s)coloring of G[P ] in which each component is scolored extends to an (r + s)coloring of G. If G does not contract to K 5 or is planar and s 2, then any (r + s 1)coloring of P in w ..."
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Cited by 5 (1 self)
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Suppose G is rcolorable and P V (G) is such that the components of G[P ] are far apart. We show that any (r + s)coloring of G[P ] in which each component is scolored extends to an (r + s)coloring of G. If G does not contract to K 5 or is planar and s 2, then any (r + s 1)coloring of P in which each component is scolored extends to an (r + s 1)coloring of G. This result uses the Four Color Theorem and its equivalence to Hadwiger's Conjecture for k = 5. For s = 2 this provides an armative answer to a question of Thomassen. Similar results hold for coloring arbitrary graphs embedded in both orientable and nonorientable surfaces.
Recent Excluded Minor Theorems
 Surveys in Combinatorics, LMS Lecture Note Series
"... We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3coloring conjecture, and Pfaffian orientations of bipartite graphs. ..."
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Cited by 3 (1 self)
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We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3coloring conjecture, and Pfaffian orientations of bipartite graphs.
Packing seagulls
, 2009
"... A seagull in a graph is an induced threevertex path. When does a graph G have k pairwise vertexdisjoint seagulls? This is NPcomplete in general, but for graphs with no stable set of size three we give a complete solution. This case is of special interest because of a connection with Hadwiger’s con ..."
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Cited by 3 (1 self)
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A seagull in a graph is an induced threevertex path. When does a graph G have k pairwise vertexdisjoint seagulls? This is NPcomplete in general, but for graphs with no stable set of size three we give a complete solution. This case is of special interest because of a connection with Hadwiger’s conjecture which was the motivation for this research; and we deduce a unification and strengthening of two theorems of Blasiak [2] concerned with Hadwiger’s conjecture. Our main result is that a graph G (different from the fivewheel) with no threevertex stable set contains k disjoint seagulls if and only if • V (G)  ≥ 3k • G is kconnected, • for every clique C of G, if D denotes the set of vertices in V (G) \C that have both a neighbour and a nonneighbour in C then D  + V (G) \ C  ≥ 2k, and • the complement graph of G has a matching with k edges. We also address the analogous fractional and halfintegral packing questions, and give a polynomial time algorithm to test whether there are k disjoint seagulls.