Circular $(4-ε)$-coloring of some classes of signed graphs

Abstract

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 graph Ĝ 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.

Signed bipartite circular cliques and a bipartite analogue of Grötzsch's theorem

“…Indeed that is the case for (K 4,4 , M ): it bounds the class of all signed bipartite planar simple graphs [20]. As the limit of the circular chromatic number of signed bipartite planar simple graphs is 4 (see [23] and [12]), this cannot be the It is shown in [19] that every signed bipartite planar graph of negative girth at least 8 maps to C −4 and that this girth condition cannot be improved to 6. We observe that C −4 is a subgraph of each of the three homomorphism targets of this discussion.…”

“…Thus for an odd integer k, and after suitable switchings, P * k consists of all planar graphs of odd girth at least k with all edges being assigned positive signs. For an even value of k, the class P * k consists of all signed planar bipartite graphs of negative girth at least k. A main question then is to find χ c (P * k ) for each k. For k = 3 and 4 both answers are 4, the first by the 4-Color Theorem, the second by the observation that 4 is the upper bound for the class of signed bipartite simple graphs and a construction given in [23] showing that 4 cannot be improved (see also [12]). For k = 5, we have the Grötzsch theorem, that gives upper bound of 3 which is also shown to be the optimal value.…”

“…In this table, when we write χ c (C) = r, it means that χ c ( Ĝ) ≤ r for each member Ĝ of the class C and that the equality is known to hold for at least one member of the class. When we write χ c (C) ∼ = r, we mean that there is a sequence of signed graphs of C whose limit of the circular chromatic number is r. In such cases, sometimes it is verified that the r is never reached by a single member of C. For example, this is indeed the case for P * 4 as shown in [12]. In other cases, it is not known if the equality holds for some members or the inequality is always strict.…”