Coloring Planar Graphs via Colored Paths in the Associahedra

Hassler Whitney's theorem of 1931 reduces the task of finding proper, vertex 4-colorings of triangulations of the 2-sphere to finding such colorings for the class $\mathfrak H$ of triangulations of the 2-sphere that have a Hamiltonian circuit. This has been used by Whitney and others from 1936 to the present to find equivalent reformulations of the 4 Color Theorem (4CT). Recently there has been activity to try to use some of these reformuations to find a shorter proof of the 4CT. Every triangulation in $\mathfrak H$ has a dual graph that is a union of two binary trees with the same number of leaves. Elements of a group known as Thompson's group $F$ are equivalence classes of pairs of binary trees with the same number of leaves. This paper explores this resemblance and finds that some recent reformulations of the 4CT are essentially attempting to color elements of $\mathfrak H$ using expressions of elements of $F$ as words in a certain generating set for $F$. From this, we derive information about not just the colorability of certain elements of $\mathfrak H$, but also about all possible ways to color these elements. Because of this we raise (and answer some) questions about enumeration. We also bring in an extension $E$ of the group $F$ and ask whether certain elements ``parametrize'' the set of all colorings of the elements of $\mathfrak H$ that use all four colors.Comment: 74 pages, table of contents, index. Revision of V.

Weighted Graph Colorings

We study two weighted graph coloring problems, in which one assigns $q$ colors to the vertices of a graph such that adjacent vertices have different colors, with a vertex weighting $w$ that either disfavors or favors a given color. We exhibit a weighted chromatic polynomial $Ph(G,q,w)$ associated with this problem that generalizes the chromatic polynomial $P(G,q)$. General properties of this polynomial are proved, and illustrative calculations for various families of graphs are presented. We show that the weighted chromatic polynomial is able to distinguish between certain graphs that yield the same chromatic polynomial. We give a general structural formula for $Ph(G,q,w)$ for lattice strip graphs $G$ with periodic longitudinal boundary conditions. The zeros of $Ph(G,q,w)$ in the $q$ and $w$ planes and their accumulation sets in the limit of infinitely many vertices of $G$ are analyzed. Finally, some related weighted graph coloring problems are mentioned.Comment: 60 pages, 6 figure

On the numbers of 1-factors and 1-factorizations of hypergraphs

A 1-factor of a hypergraph $G=(X,W)$ is a set of hyperedges such that every vertex of $G$ is incident to exactly one hyperedge from the set. A 1-factorization is a partition of all hyperedges of $G$ into disjoint 1-factors. The adjacency matrix of a $d$-uniform hypergraph $G$ is the $d$-dimensional (0,1)-matrix of order $|X|$ such that an element $a_{\alpha_1, \ldots, \alpha_d}$ of $A$ equals 1 if and only if $\left\{\alpha_1, \ldots, \alpha_d\right\}$ is a hyperedge of $G$. Here we estimate the number of 1-factors of uniform hypergraphs and the number of 1-factorizations of complete uniform hypergraphs by means of permanents of their adjacency matrices

Tutte Polynomials and Related Asymptotic Limiting Functions for Recursive Families of Graphs

We prove several theorems concerning Tutte polynomials $T(G,x,y)$ for recursive families of graphs. In addition to its interest in mathematics, the Tutte polynomial is equivalent to an important function in statistical physics, the Potts model partition function of the $q$-state Potts model, $Z(G,q,v)$, where $v$ is a temperature-dependent variable. We determine the structure of the Tutte polynomial for a cyclic clan graph $G[(K_r)_m,L=jn]$ comprised of a chain of $m$ copies of the complete graph $K_r$ such that the linkage $L$ between each successive pair of $K_r$'s is a join $jn$, and $r$ and $m$ are arbitrary. The explicit calculation of the case $r=3$ (for arbitrary $m$) is presented. The continuous accumulation set of the zeros of $Z$ in the limit $m \to \infty$ is considered. Further, we present calculations of two special cases of Tutte polynomials, namely, flow and reliability polynomials, for cyclic clan graphs and discuss the respective continuous accumulation sets of their zeros in the limit $m \to \infty$. Special valuations of Tutte polynomials give enumerations of spanning trees and acyclic orientations. Two theorems are presented that determine the number of spanning trees on $G[(K_r)_m,jn]$ and $G[(K_r)_m,id]$, where $L=id$ means that the identity linkage. We report calculations of the number of acyclic orientations for strips of the square lattice and use these to obtain an improved lower bound on the exponential growth rate of the number of these acyclic orientations.Comment: 50 pages, latex, 5 figure

From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group

We show that correlators of the hermitian one-Matrix model with a general potential can be mapped to the counting of certain triples of permutations and hence to counting of holomorphic maps from world-sheet to sphere target with three branch points on the target. This allows the use of old matrix model results to derive new explicit formulae for a class of Hurwitz numbers. Holomorphic maps with three branch points are related, by Belyi's theorem, to curves and maps defined over algebraic numbers \bmQ. This shows that the string theory dual of the one-matrix model at generic couplings has worldsheets defined over the algebraic numbers and a target space \mP^1 (\bmQ). The absolute Galois group Gal (\bmQ / \mQ) acts on the Feynman diagrams of the 1-matrix model, which are related to Grothendieck's Dessins d'Enfants. Correlators of multi-matrix models are mapped to the counting of triples of permutations subject to equivalences defined by subgroups of the permutation groups. This is related to colorings of the edges of the Grothendieck Dessins. The colored-edge Dessins are useful as a tool for describing some known invariants of the Gal (\bmQ / \mQ) action on Grothendieck Dessins and for defining new invariants.Comment: 54 pages, 17 figure

Knot theory for self-indexed graphs

We introduce and study so-called self-indexed graphs. These are (oriented) finite graphs endowed with a map from the set of edges to the set of vertices. Such graphs naturally arise from classical knot and link diagrams. In fact, the graphs resulting from link diagrams have an additional structure, an integral flow. We call a self-indexed graph with integral flow a comte. The analogy with links allows us to define transformations of comtes generalizing the Reidemeister moves on link diagrams. We show that many invariants of links can be generalized to comtes, most notably the linking number, the Alexander polynomials, the link group, etc. We also discuss finite type invariants and quandle cocycle invariants of comtes.Comment: 14 page

A 66-chromatic two-distance graph in the plane

We prove that if one colors each point of the Euclidean plane with one of five colors, then there exist two points of the same color that are either distance $1$ or distance $2$ apart.Comment: 7 pages, 1 figur

An analytic family of representations for the mapping class group of punctured surfaces

We use quantum invariants to define an analytic family of representations for the mapping class group of a punctured surface. The representations depend on a complex number A with |A| <= 1 and act on an infinite-dimensional Hilbert space. They are unitary when A is real or imaginary, bounded when |A|<1, and only densely defined when |A| = 1 and A is not a root of unity. When A is a root of unity distinct from 1, -1, i, -i the representations are finite-dimensional and isomorphic to the "Hom" version of the well-known TQFT quantum representations. The unitary representations in the interval [-1,0] interpolate analytically between two natural geometric unitary representations, the SU(2)-character variety representation studied by Goldman and the multicurve representation induced by the action of the mapping class group on multicurves. The finite-dimensional representations converge analytically to the infinite-dimensional ones. We recover Marche and Narimannejad's convergence theorem, and Andersen, Freedman, Walker and Wang's asymptotic faithfulness, that states that the image of a non-central mapping class is always non-trivial after some level r. When the mapping class is pseudo-Anosov we give a simple polynomial estimate of the level r in term of its dilatation.Comment: 41 pages, 13 figure

Square numbers, spanning trees and invariants of achiral knots

We give constructions to realize an odd number, which is representable as sum of two squares, as determinant of an achiral knot, thus proving that these are exactly the numbers occurring as such determinants. Later we study which numbers occur as determinants of prime alternating achiral knots, and obtain a complete result for perfect squares. Using the checkerboard coloring, then an application is given to the number of spanning trees in planar self-dual graphs. Another application are some enumeration results on achiral rational knots. Finally, we describe the leading coefficients of the Alexander and skein polynomial of alternating achiral knots.Comment: 22 pages, 4 figures; revision 2 Sep 01: some results extended and questions answered, new examples included, application to spanning trees added, Title changed; revision 26 Feb 03: several sections reorganized and slightly extended, in last section citations and proofs for more special cases of the questions adde

Trace functionals on non-commutative deformations of moduli spaces of flat connections

We describe an efficient construction of a canonical non-commutative deformation of the algebraic functions on the moduli spaces of flat connections on a Riemann surface. We show that this algebra, which is a variant of the quantum moduli algebra introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche, has a trace functional which is related to the canonical trace in the formal index theory of Fedosov and Nest-Tsygan via the Verlinde formula