Signed graph coloring

Coloring signed graphs using DFS

“…The concept of signed graphs is due to Harary [2]. Vertex-colorings of signed graphs were introduced by Zaslavsky [5] in the following way. Let G be a signed graph and r ≥ 0 an integer, now a function c : V (G) → {−r, −r + 1, .…”

A Bivariate Chromatic Polynomial for Signed Graphs

“…This bivariate chromatic polynomial (in Corollary 5 we will see that c (2k + 1, 2l) is indeed a polynomial) specializes to Zaslavsky's chromatic polynomial of signed graphs [11] in the case l = 0. As in Zaslavsky's theory, c (2k + 1, 2l) comes with a companion, the zero-free bivariate chromatic polynomial c * (2k, 2l) which counts all…”

“…Zaslavsky [11] proved the following analogue of Stanley's Theorem 3 for the chromatic polynomial c (2k + 1) of a signed graph : Theorem 9 (Zaslavsky) For k ∈ Z >0 , (−1) |V | c (−2k − 1) equals the number of k-colorings of , each counted with multiplicity equal to the number of compatible acyclic orientations of . In particular, (−1) |V | c (−1) equals the number of acyclic orientations of .…”

“…Following [11], we call a coloring x ∈ Z V and an orientation η compatible if for any link e = vw η ve x v + η we x w ≥ 0 , and for any halfedge or negative loop e at v η ve x v ≥ 0 .…”

A Factorization Theorem of Characteristic Polynomials of Convex Geometries

“…(1) signed graph coloring theory by Zaslavsky [13][14][15] for the hyperplane arrangements contained in the Weyl arrangements B n , (2) freeness of hyperplane arrangements by Saito [10] and by Terao [12], (3) supersolvability of upper semimodular lattices by Stanley [11].…”