Polynomial graph-colorings

Curvature from Graph Colorings

Given a finite simple graphG = (V,E) with chromatic number c and chromatic polynomial C(x). Every vertex graph coloring f of G defines an index if (x) satisfying the Poincaré-Hopf theorem [17] ∑ x if (x) = χ(G). As a variant to the index expectation result [19] we prove that E[if (x)] is equal to curvature K(x) satisfying Gauss-Bonnet ∑ xK(x) = χ(G) [16], where the expectation is the average ov...

Reconfiguring Graph Colorings

Graph coloring has been studied for a long time and continues to receive interest within the research community [43]. It has applications in scheduling [46], timetables, and compiler register allocation [45]. The most popular variant of graph coloring, k-coloring, can be thought of as an assignment of k colors to the vertices of a graph such that adjacent vertices are assigned different colors....

Acyclic Colorings of Graph Subdivisions

An acyclic coloring of a graph G is a coloring of the vertices of G, where no two adjacent vertices of G receive the same color and no cycle of G is bichromatic. An acyclic k-coloring of G is an acyclic coloring of G using at most k colors. In this paper we prove that any triangulated plane graph G with n vertices has a subdivision that is acyclically 4-colorable, where the number of division v...

Fibonacci Identities and Graph Colorings

We generalize both the Fibonacci and Lucas numbers to the context of graph colorings, and prove some identities involving these numbers. As a corollary we obtain new proofs of some known identities involving Fibonacci numbers such as Fr+s+t = Fr+1Fs+1Ft+1 + FrFsFt − Fr−1Fs−1Ft−1.

On Szeged Polynomial of a Graph