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13
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 34 (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.
The optimal pathmatching problem
 COMBINATORICA
, 1997
"... We describe a common generalization of the weighted matching problem and the weighted matroid intersection problem. In this context we establish common generalizations of the main results on those two problemspolynomialtime solvability, minmax theorems, and totally dual integral polyhedral descr ..."
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Cited by 24 (2 self)
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We describe a common generalization of the weighted matching problem and the weighted matroid intersection problem. In this context we establish common generalizations of the main results on those two problemspolynomialtime solvability, minmax theorems, and totally dual integral polyhedral descriptions. New applications of these results include a strongly polynomial separation algorithm for the convex hull of matchable sets of a graph, and a polynomialtime algorithm to compute the rank of a certain matrix of indeterminates.
Solving connectivity problems parameterized by treewidth in single exponential time (Extended Abstract)
, 2011
"... For the vast majority of local problems on graphs of small treewidth (where by local we mean that a solution can be verified by checking separately the neighbourhood of each vertex), standard dynamic programming techniques give c tw V  O(1) time algorithms, where tw is the treewidth of the input g ..."
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Cited by 12 (1 self)
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For the vast majority of local problems on graphs of small treewidth (where by local we mean that a solution can be verified by checking separately the neighbourhood of each vertex), standard dynamic programming techniques give c tw V  O(1) time algorithms, where tw is the treewidth of the input graph G = (V, E) and c is a constant. On the other hand, for problems with a global requirement (usually connectivity) the best–known algorithms were naive dynamic programming schemes running in at least tw tw time. We breach this gap by introducing a technique we named Cut&Count that allows to produce c tw V  O(1) time Monte Carlo algorithms for most connectivitytype problems, including HAMILTONIAN PATH, STEINER TREE, FEEDBACK VERTEX SET and CONNECTED DOMINATING SET. These results have numerous consequences in various fields, like parameterized complexity, exact and approximate algorithms on planar and Hminorfree graphs and exact algorithms on graphs of bounded degree. The constant c in our algorithms is in all cases small, and in several cases we are able to show that improving those constants would cause the Strong Exponential Time Hypothesis to fail. In contrast to the problems aiming to minimize the number of connected components that we solve using Cut&Count as mentioned above, we show that, assuming the Exponential Time Hypothesis, the aforementioned gap cannot be breached for some problems that aim to maximize the number of connected components like CYCLE PACKING.
An algebraic matching algorithm
 COMBINATORICA 20 (1) (2000) 61–70
, 2000
"... Tutte introduced a V by V skewsymmetric matrix T =(tij), called the Tutte matrix, associated witha simple graph G =(V,E). He associates an indeterminate ze with each e ∈ E, then defines tij = ±ze when ij = e ∈ E, and tij = 0 otherwise. The rank of the Tutte matrix is exactly twice the size of a max ..."
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Cited by 7 (2 self)
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Tutte introduced a V by V skewsymmetric matrix T =(tij), called the Tutte matrix, associated witha simple graph G =(V,E). He associates an indeterminate ze with each e ∈ E, then defines tij = ±ze when ij = e ∈ E, and tij = 0 otherwise. The rank of the Tutte matrix is exactly twice the size of a maximum matching of G. Using linear algebra and ideas from the Gallai–Edmonds decomposition, we describe a very simple yet efficient algorithm that replaces the indeterminates with constants without losing rank. Hence, by computing the rank of the resulting matrix, we can efficiently compute the size of a maximum matching of a graph.
The circular chromatic index of graphs of high girth
 J. COMBIN. TH. (B
"... We show that for each ε>0 and each integer ∆ ≥ 1, there exists a number g such that for any graph G of maximum degree ∆ and girth at least g, the circular chromatic index of G is at most ∆ + ε. ..."
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Cited by 5 (2 self)
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We show that for each ε>0 and each integer ∆ ≥ 1, there exists a number g such that for any graph G of maximum degree ∆ and girth at least g, the circular chromatic index of G is at most ∆ + ε.
Clique Minors In Graphs And Their Complements
, 2000
"... A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. Let t ≥ 1 be an integer, and let G be a graph on n vertices with no minor isomorphic to Kt+1. Kostochka conjectures that there exists a constant c = c(k) independent of G such that the complement of G h ..."
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Cited by 1 (0 self)
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A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. Let t ≥ 1 be an integer, and let G be a graph on n vertices with no minor isomorphic to Kt+1. Kostochka conjectures that there exists a constant c = c(k) independent of G such that the complement of G has a minor isomorphic to Ks, wheres=⌈1 2 (1 + 1/t)n − c⌉. We prove that Kostochka’s conjecture is equivalent to the conjecture of Duchet and Meyniel that every graph with no minor isomorphic to Kt+1 has an independent set of size at least n/t. We deduce that Kostochka’s conjecture holds for all integers t ≤ 5, and that a weaker form with s replaced by s ′ = ⌈ 1 2 (1 + 1/(2t))n − c⌉ holds for all integers t ≥ 1.
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"... In a series of papers (see [1], [2], [3]) we have considered the structure of a random graph T,,, N obtained as follows: we select at random N edges among the n ( n possible edges connecting n given points so that each of the ..."
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In a series of papers (see [1], [2], [3]) we have considered the structure of a random graph T,,, N obtained as follows: we select at random N edges among the n ( n possible edges connecting n given points so that each of the
Matching, Matroids, and Extensions
, 2001
"... Perhaps the two most fundamental wellsolved models in combinatorial optimization are the optimal matching problem and the optimal matroid intersection problem. We review the basic results for both, and describe some more recent advances. Then we discuss extensions of these models, in particular, tw ..."
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Perhaps the two most fundamental wellsolved models in combinatorial optimization are the optimal matching problem and the optimal matroid intersection problem. We review the basic results for both, and describe some more recent advances. Then we discuss extensions of these models, in particular, two recent ones  jump systems and pathmatchings.
Theorems and Algorithms
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Oxford is a registered trademark of Oxford University Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress CataloginginPublication Data