Related papers: Three-Colorings of Cubic Graphs and Tensor Operato…

We extend the Penrose polynomial, originally defined only for plane graphs, to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial of embedded graphs leads to new identities and relations for the Penrose polynomial…

A recent lower bound on the number of edges in a k-critical n-vertex graph by Kostochka and Yancey yields a half-page proof of the celebrated Gr\"otzsch Theorem that every planar triangle-free graph is 3-colorable. In this paper we use the…

We give an exact characterization of 3-colorability of triangle-free graphs drawn in the torus, in the form of 186 "templates" (graphs with certain faces filled by arbitrary quadrangulations) such that a graph from this class is not…

We give a linear-time algorithm to decide 3-colorability (and find a 3-coloring, if it exists) of quadrangulations of a fixed surface. The algorithm also allows to prescribe the coloring for a bounded number of vertices.

The Four color problem is closely related to other branches of mathematics and practical applications. More than 20 of its reformulations are known, which connect this problem with problems of algebra, statistical mechanics and planning.…

For any cubic graph in a closed orientable surface and a perfect matching, the Penrose-Kauffman polynomial is a sum of chromatic polynomials of a collection of associated graphs. A knot-theoretic perspective affords elementary proofs of old…

We prove that every cyclically 4-edge-connected cubic graph that can be embedded in the projective plane, with the single exception of the Petersen graph, is 3-edge-colorable. In other words, the only (non-trivial) snark that can be…

We prove that every cyclically 4-edge-connected cubic graph that can be embedded in the torus, with the exceptional graph class called "Petersen-like", is 3-edge-colorable. This means every (non-trivial) toroidal snark can be obtained from…

Thomassen formulated the following conjecture: Every $3$-connected cubic graph has a red-blue vertex coloring such that the blue subgraph has maximum degree $1$ (that is, it consists of a matching and some isolated vertices) and the red…

We survey work on coloring, list coloring, and painting squares of graphs; in particular, we consider strong edge-coloring. We focus primarily on planar graphs and other sparse classes of graphs.

The precoloring problem of a graph involves assigning colors to some vertices beforehand, and the objective is to determine whether it can be extended to a proper k-coloring of the entire graph. In 1958, Grotzsch proved that every…

A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induce a union of disjoint paths. In this paper, we prove that for every connected graph with maximum degree at most three and…

Thomassen conjectured that every triangle-free planar graph on n vertices has exponentially many 3-colorings, and proved that it has at least 2^[n^(1/12)/20000] distinct 3-colorings. We show that it has at least 2^sqrt(n/362) distinct…

The notion of the genus of a quadratic form is generalized to vertex operator algebras. We define it as the modular braided tensor category associated to a suitable vertex operator algebra together with the central charge. Statements…

Consider the collection of edge bicolorings of a graph that is cellularly embedded on an orientable surface. In this work, we count the number of equivalence classes of such colorings under two relations: reversing colors around a face and…

Tensor network methods are taking a central role in modern quantum physics and beyond. They can provide an efficient approximation to certain classes of quantum states, and the associated graphical language makes it easy to describe and…

Anchored planar algebras, a generalized notion of Vaughan Jones' planar algebras, have recently seen use in higher category theory, functional analysis, and TQFT applications. These algebras are equipped with a natural 3-dimensional…

It is known that every loopless cubic graph is 4-edge choosable. We prove the following strengthened result. Let G be a planar cubic graph having b cut-edges. There exists a set F of at most 5b/2 edges of G with the following property. For…

This paper presents infinitely many new examples of triangle-free uniquely 3-edge colorable cubic graphs. The only such graph previously known was given by Tutte in 1976.

In a proper edge-coloring of a cubic graph, an edge $e$ is normal if the set of colors used by the edges adjacent to $e$ has cardinality 3 or 5. The Petersen coloring conjecture asserts that every bridgeless cubic graph has a normal…