Circular (4 − \epsilon)-coloring of some classes of signed graphs
Résumé
A circular r-coloring of a signed graph (G, σ) is an assignment φ of points of a circle C r of circumference r to the vertices of (G, σ) such that for each positive edge uv of (G, σ) the distance of φ(v) and φ(v) is at least 1 and for each negative edge uv the distance of φ(u) from the antipodal of φ(v) is at least 1. The circular chromatic number of (G, σ), denoted χ c (G, σ), is the infimum of r such that (G, σ) admits a circular r-coloring. This notion is recently defined by Naserasr, Wang, and Zhu who, among other results, proved that for any signed d-degenerate simple graphĜ we have χ c (Ĝ) ≤ 2d. For d ≥ 3, examples of signed d-degenerate simple graphs of circular chromatic number 2d are provided. But for d = 2 only examples of signed 2-degenerate simple graphs of circular chromatic number close enough to 4 are given, noting that these examples are also signed bipartite planar graphs. In this work we first observe the following restatement of the 4-color theorem: If (G, σ) is a signed bipartite planar simple graph where vertices of one part are all of degree 2, then χ c (G, σ) ≤ 16 5. Motivated by this observation, we provide an improved upper bound of 4 − 2 n+1 2 for the circular chromatic number of a signed 2-degenerate simple graph on n vertices and an improved upper bound of 4 − 4 n+2 2 for the circular chromatic number of a signed bipartite planar simple graph on n vertices. We then show that each of the bounds is tight for any value of n ≥ 4.
Domaines
Mathématique discrète [cs.DM]
Fichier principal
Circular4ColoringSignedBipartitePlanarGraphs_July25.pdf (330.32 Ko)
Télécharger le fichier
Origine | Fichiers produits par l'(les) auteur(s) |
---|