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A simpler linear time algorithm for embedding graphs into an arbitrary surface and the genus of graphs of bounded treewidth
, 2008
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Approximation algorithms via contraction decomposition
 Proc. 18th Ann. ACMSIAM Symp. Discrete Algorithms ACMSIAM symposium on Discrete algorithms
, 2007
"... We prove that the edges of every graph of bounded (Euler) genus can be partitioned into any prescribed number k of pieces such that contracting any piece results in a graph of bounded treewidth (where the bound depends on k). This decomposition result parallels an analogous, simpler result for edge ..."
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Cited by 41 (8 self)
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We prove that the edges of every graph of bounded (Euler) genus can be partitioned into any prescribed number k of pieces such that contracting any piece results in a graph of bounded treewidth (where the bound depends on k). This decomposition result parallels an analogous, simpler result for edge deletions instead of contractions, obtained in [Bak94, Epp00, DDO + 04, DHK05], and it generalizes a similar result for “compression ” (a variant of contraction) in planar graphs [Kle05]. Our decomposition result is a powerful tool for obtaining PTASs for contractionclosed problems (whose optimal solution only improves under contraction), a much more general class than minorclosed problems. We prove that any contractionclosed problem satisfying just a few simple conditions has a PTAS in boundedgenus graphs. In particular, our framework yields PTASs for the weighted Traveling Salesman Problem and for minimumweight cedgeconnected submultigraph on boundedgenus graphs, improving and generalizing previous algorithms of [GKP95, AGK + 98, Kle05, Gri00, CGSZ04, BCGZ05]. We also highlight the only main difficulty in extending our results to general Hminorfree graphs.
Graph Minor Theory
 BULLETIN (NEW SERIES) OF THE AMERICAN MATHEMATICAL SOCIETY
, 2005
"... A monumental project in graph theory was recently completed. The project, started by Robertson and Seymour, and later joined by Thomas, led to entirely new concepts and a new way of looking at graph theory. The motivating problem was Kuratowski’s characterization of planar graphs, and a farreaching ..."
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Cited by 22 (0 self)
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A monumental project in graph theory was recently completed. The project, started by Robertson and Seymour, and later joined by Thomas, led to entirely new concepts and a new way of looking at graph theory. The motivating problem was Kuratowski’s characterization of planar graphs, and a farreaching generalization of this, conjectured by Wagner: If a class of graphs is minorclosed (i.e., it is closed under deleting and contracting edges), then it can be characterized by a finite number of excluded minors. The proof of this conjecture is based on a very general theorem about the structure of large graphs: If a minorclosed class of graphs does not contain all graphs, then every graph in it is glued together in a treelike fashion from graphs that can almost be embedded in a fixed surface. We describe the precise formulation of the main results and survey some of its applications to algorithmic and structural problems in graph theory.
Coloringflow duality of embedded graphs
 TRANS. AMER. MATH. SOC
, 2004
"... Let G be a directed graph embedded in a surface. A map φ: E(G) → R is a tension if for every circuit C ⊆ G, thesumofφon the forward edges of C is equal to the sum of φ on the backward edges of C. If this condition is satisfied for every circuit of G which is a contractible curve in the surface, the ..."
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Cited by 6 (3 self)
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Let G be a directed graph embedded in a surface. A map φ: E(G) → R is a tension if for every circuit C ⊆ G, thesumofφon the forward edges of C is equal to the sum of φ on the backward edges of C. If this condition is satisfied for every circuit of G which is a contractible curve in the surface, then φ is a local tension. If 1 ≤φ(e)  ≤α − 1holdsforevery e ∈ E(G), we say that φ is a (local) αtension. We define the circular chromatic number and the local circular chromatic number of G by χc(G) =inf{α ∈ R  G has an αtension} and χloc(G) =inf{α∈R  G has a local αtension}, respectively. The invariant χc is a refinement of the usual chromatic number, whereas χloc is closely related to Tutte’s flow index and Bouchet’s biflow index of the surface dual G ∗. From the definitions we have χloc(G) ≤ χc(G). The main result of this paper is a farreaching generalization of Tutte’s coloringflow duality in planar graphs. It is proved that for every surface X and every ε>0, there exists an integer M so that χc(G) ≤ χloc(G)+ε holds for every graph embedded in X with edgewidth at least M, where the edgewidth is the length of a shortest noncontractible circuit in G. In 1996, Youngs discovered that every quadrangulation of the projective plane has chromatic number 2 or 4, but never 3. As an application of the main result we show that such ‘bimodal ’ behavior can be observed in χloc, and thus in χc for two generic classes of embedded graphs: those that are triangulations and those whose face boundaries all have even length. In particular, if G is embedded in some surface with large edgewidth and all its faces have even length ≤ 2r, thenχc(G) ∈ [2, 2+ε] ∪ [ 2r, 4]. Similarly, if G is a triangulation r−1 with large edgewidth, then χc(G) ∈ [3, 3+ε]∪[4, 5]. It is also shown that there exist Eulerian triangulations of arbitrarily large edgewidth on nonorientable surfaces whose circular chromatic number is equal to 5.
A SPECIAL CASE OF HADWIGER’S CONJECTURE
, 2005
"... Abstract. We investigate Hadwiger’s conjecture for graphs with no stable set of size 3. Such a graph on at least 2t − 1 vertices is not t − 1 colorable, so is conjectured to have a Kt minor. There is a strengthening of Hadwiger’s conjecture in this case, which states that there is always a minor in ..."
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Abstract. We investigate Hadwiger’s conjecture for graphs with no stable set of size 3. Such a graph on at least 2t − 1 vertices is not t − 1 colorable, so is conjectured to have a Kt minor. There is a strengthening of Hadwiger’s conjecture in this case, which states that there is always a minor in which the preimages of the vertices of Kt are connected subgraphs of size one or two. We prove this strengthened version for graphs whose complement has an even number of vertices and fractional chromatic number less than 3. We investigate several possible generalizations and obtain counterexamples for some and improved results from others. We also show that for sufficiently large n = V (G), a graph with no stable set of size 3 has a K 1 9 n4/5 minor using only sets of size one or two as preimages of vertices. 1.
Circular chromatic number of evenfaced projective plane graphs
"... We derive an exact formula for the circular chromatic number of any graph embeddable on the projective plane in such a way that all of it faces have even length. ..."
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We derive an exact formula for the circular chromatic number of any graph embeddable on the projective plane in such a way that all of it faces have even length.
New Methods in Celestial
"... Abstract. Alongstanding and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. ..."
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Abstract. Alongstanding and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any integer k greater than or equal to 3, there exist infinitely many arithmetic progressions of length k consisting only of prime numbers. This is an introduction to some of the ideas in the proof, concentrating on those drawn from ergodic theory. 1. Background Forhundreds of years, mathematicians have made conjectures about patterns in the primes: one of the simplest to state is that the primes contain arbitrarily long arithmetic progressions. It is not clear exactly