In a seminal paper, De Loera et. al introduce the algorithm NulLA (Nullstellensatz Linear Algebra) and use it to measure the difficulty of determining if a graph is not 3-colorable. The crux of this relies on a correspondence between 3-colorings of a graph and solutions to a certain system of polynomial equations over a field k. In this article, we give a new direct combinatorial characterizati...

Graph coloring in general is an extremely easy-to-understand yet powerful tool. It has wide-ranging applications from register allocation to image segmentation. For such a simple problem, however, the question is surprisingly intractable. In this section I will introduce the problem formally, as well as present some general background on graph coloring. There are several ways to color a graph, ...

We study the maximization version of the fundamental graph coloring problem. Here the goal is to color the vertices of a k-colorable graph with k colors so that a maximum fraction of edges are properly colored (i.e. their endpoints receive different colors). A random k-coloring properly colors an expected fraction 1 − 1 k of edges. We prove that given a graph promised to be k-colorable, it is N...

For an integer k 2 2, a proper k-restraint on a graph G is a function from the vertex set of G to the set of k-colors. A graph G is amenably k-colorable if, for each nonconstant proper k-restraint r on G, there is a k-coloring c of G with c(v) # r(v) for each vertex v of G. A graph G is amenable if it is amenably k-colorable and k is the chromatic number of G. For any k Z= 3, there are infinite...

Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colorable if for every partition of V (G) into disjoint sets V1 ∪ . . . ∪ Vr, all of size exactly k, there exists a proper vertex k-coloring of G with each color appearing exactly once in each Vi. In the case when k does not divide n, G is defined to be strongly k-colorable if the graph obtained by ...

Let G = (V,E) be a multigraph of maximum degree ∆. The edges of G can be colored with at most 3 2∆ colors by Shannon’s theorem. We study lower bounds on the size of subgraphs of G that can be colored with ∆ colors. Shannon’s Theorem gives a bound of ∆ ⌊ 3 2 ∆⌋ |E|. However, for ∆ = 3, Kamiński and Kowalik [7, 8] showed that there is a 3-edge-colorable subgraph of size at least 79 |E|, unless G ...

In this paper, we prove that planar graphs without cycles of length 4, 6, 9 are 3-colorable.

A simple characterization of the 3, 4, or 5-colorable Eulerian triangulations of the projective plane is given.

Bn : (∀x1, . . . , xn)( if all xi are distinct then the subgraph induced on x1, . . . , xn is 3-colorable). By Erdős-DeBruijn, the countable set of axioms Bn (n = 1, 2, . . . ) defines 3-colorability. To show that 3-colorability is not finitely axiomatizable, we show the (apparently) stronger result that non-3-colorability is not axiomatizable. To do this, we use ultraproducts. It suffices to c...

A $(d,h)$-decomposition of a graph $G$ is an ordered pair $(D, H)$ such that $H$ subgraph maximum degree at most $h$ and $D$ acyclic orientation $G-E(H)$ with out-degree $d$. In this paper, we prove for $l \in \{5, 6, 7, 8, 9\}$, every planar without $4$- $l$-cycles $(2,1)$-decomposable. As consequence, $l$-cycles, there exists matching $M$, $G - M$ $3$-DP-colorable has Alon-Tarsi number $3$. p...