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14
A Tutte polynomial for signed graphs
 Discrete Appl. Math
, 1989
"... This paper introduces a generalization of the Tutte polynomial [14] that is defined for signed graphs. A signed graph is a graph whose edges are each labelled with a sign (+l or 1). The generalized polynomial will be denoted Q[G] = Q[G](A, B, d). Here G is the signed graph, and the letters A, B, d ..."
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This paper introduces a generalization of the Tutte polynomial [14] that is defined for signed graphs. A signed graph is a graph whose edges are each labelled with a sign (+l or 1). The generalized polynomial will be denoted Q[G] = Q[G](A, B, d). Here G is the signed graph, and the letters A, B, d denote three independent
Some probabilistic restatements of the four color conjecture
 Journal of Graph Theory
, 2004
"... With every triangulation of sphere we associate in a natural way a probabilistic space and define several random events. The Four Color Conjecture turns out to be equivalent to different statements about positive correlation among some pairs of these events. c ○ (2003) John Wiley & Sons, Inc. 1. ..."
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With every triangulation of sphere we associate in a natural way a probabilistic space and define several random events. The Four Color Conjecture turns out to be equivalent to different statements about positive correlation among some pairs of these events. c ○ (2003) John Wiley & Sons, Inc. 1.
The Algebra of 3Graphs
 Proc. Steklov Inst. Math. 221
, 1998
"... We introduce and study the structure of an algebra in the linear space spanned by all regular 3valent graphs with a prescribed order of edges at every vertex, modulo certain relations. The role of this object in various areas of low dimensional topology is discussed. 0 Introduction Regular gra ..."
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We introduce and study the structure of an algebra in the linear space spanned by all regular 3valent graphs with a prescribed order of edges at every vertex, modulo certain relations. The role of this object in various areas of low dimensional topology is discussed. 0 Introduction Regular graphs of degree 3, i. e. graphs in which every vertex is incident with exactly three edges, often occur in mathematics. Apart from graph theory proper, where such graphs are referred to as `cubic', they appear in a natural way in the topology of 3manifolds, in the Vassiliev knot invariant theory and in connection with the four colour theorem. It turns out that in all these applications 3valent graphs are endowed with a natural structure that consists in fixing, at every vertex of the graph, one of the two possible cyclic orders in the set of three edges issuing from this vertex. In the theory of Vassiliev invariants 3valent graphs can be viewed as elements of the primitive subspace P of th...
One Probabilistic Equivalent of the Four Color Conjecture
, 2003
"... For every twoconnected planar threevalent graph we introduce in a natural way a probabilistic space and define two random events; the Four Color Conjecture turns out to be eqivalent to (positive) correlation of these events. ..."
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For every twoconnected planar threevalent graph we introduce in a natural way a probabilistic space and define two random events; the Four Color Conjecture turns out to be eqivalent to (positive) correlation of these events.
TemperleyLieb Algebras And The FourColor Theorem
"... The TemperleyLieb algebra Tn with parameter 2 is the associative algebra over Q generated by 1, e0, el,..., en, where the generators satisfy the rela 2 = 2el, eiejei = ei if [i j[ = 1 and eiej = ejei if [i j[ > 2. We tions i  use the Four Color Theorem to give a necessary and sufficient con ..."
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The TemperleyLieb algebra Tn with parameter 2 is the associative algebra over Q generated by 1, e0, el,..., en, where the generators satisfy the rela 2 = 2el, eiejei = ei if [i j[ = 1 and eiej = ejei if [i j[ > 2. We tions i  use the Four Color Theorem to give a necessary and sufficient condition for certain elements of Tn to be nonzero. It turns out that the characterization is, in fact, equivalent to the Four Color Theorem.
www.elsevier.com/locate/disc Reformulating the map color theorem
, 2004
"... This paper discusses reformulations of the problem of coloring plane maps with four colors. We include discussion of the Eliahou–Kryuchkov conjecture, the Penrose formula, the vector crossproduct formulation and the reformulations in terms of formations and factorizations due to G. SpencerBrown. © ..."
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This paper discusses reformulations of the problem of coloring plane maps with four colors. We include discussion of the Eliahou–Kryuchkov conjecture, the Penrose formula, the vector crossproduct formulation and the reformulations in terms of formations and factorizations due to G. SpencerBrown. © 2005 Elsevier B.V. All rights reserved.
Graph Planarity and Related Topics
 GRAPH DRAWING (PROC. GD ’99)
, 1999
"... This compendium is the result of reformatting and minor editing of the author's transparencies for his talk delivered at the conference. The talk covered Euler's Formula, Kuratowski's Theorem, linear planarity tests, Schnyder's Theorem and drawing on the grid, the two paths problem, Pfaffian ori ..."
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This compendium is the result of reformatting and minor editing of the author's transparencies for his talk delivered at the conference. The talk covered Euler's Formula, Kuratowski's Theorem, linear planarity tests, Schnyder's Theorem and drawing on the grid, the two paths problem, Pfaffian orientations, linkless embeddings, and the Four Color Theorem.
THE COMB POSET AND THE PARSEWORDS FUNCTION
"... Abstract. In this paper we explore some of the properties of the comb poset, whose notion was first introduced by J. M. Pallo. We show that three binary functions that are not wellbehaved in the Tamari lattice are remarkably wellbehaved within an interval of the comb poset: rotation distance, meet ..."
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Abstract. In this paper we explore some of the properties of the comb poset, whose notion was first introduced by J. M. Pallo. We show that three binary functions that are not wellbehaved in the Tamari lattice are remarkably wellbehaved within an interval of the comb poset: rotation distance, meets and joins, and the common parse words function for a pair of trees. We conclude by giving explicit expressions for the number of common parse words for a pair of trees within an interval of the comb poset, a problem whose generalization is known to be equivalent to the Four Color theorem. 1.
An Approximate Restatement of the Four Color Theorem
"... The celebrated Four Color Theorem was first conjectured in the 1850’s. More than a century later Appel and Haken [1] were able to find the first proof. Previously, there had been many partial results, and many false proofs. Appel and Haken’s famous proof has one “drawback”: it makes extensive use of ..."
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The celebrated Four Color Theorem was first conjectured in the 1850’s. More than a century later Appel and Haken [1] were able to find the first proof. Previously, there had been many partial results, and many false proofs. Appel and Haken’s famous proof has one “drawback”: it makes extensive use of computer computation. More recently Robertson, Sanders, Seymour and Thomas [18] created another proof of the Four Color Theorem. However, their proof, while simplifying some technical parts of the AppelHaken proof, still relies on computer computations. There is some debate in the mathematical community about whether mathematicians should be satisfied with mathematical proofs that rely on extensive computation and there is interest in finding a new proof of the Four Color Theorem that relies on no computer computation. Such a proof would perhaps yield additional insights into why the Four Color Theorem is really true, and might yield new insights into the structure of planar graphs. In any case, there continues to be a search for such a proof. The contribution of this paper is that we initiate a new approach towards proving the Four Color Theorem. Our approach is based on insights from computer science theory and modern