Circular coloring

Abstract: A generalization of the chromatic number of a graph is introduced such that the colors are integers modulo n, and the colors on adjacent vertices are required to be as far apart as possible.

Abstract: The odd-girth of a graph is the length of a shortest odd circuit. A conjecture by Pavol Hell about circular coloring is solved in this article by showing that there is a function 𝒻(e) for each e : 0 < e < 1 such that, if the odd-girth of a planar graph G is at least 𝒻(e), then G is (2 + e)-colorable. Note that the function 𝒻(e) is independent of the graph G and e ➝ 0 if and only if 𝒻(e) ➝ ∞. A key lemma, called the folding lemma, is proved that provides a reduction method, which maintains the odd-girth of planar graphs. This lemma is expected to have applications in related problems. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 109119, 2000

Abstract: Let $k, d$ ($2d \leq k)$ be two positive integers. We generalize the well studied notions of $(k,d)$-colorings and of the circular chromatic number $\chi_c$ to signed graphs. This implies a new notion of colorings of signed graphs, and the corresponding chromatic number $\chi$. Some basic facts on circular colorings of signed graphs and on the circular chromatic number are proved, and differences to the results on unsigned graphs are analyzed. In particular, we show that the difference between the circular chromatic number and the chromatic number of a signed graph is at most 1. Indeed, there are signed graphs where the difference is 1. On the other hand, for a signed graph on $n$ vertices, if the difference is smaller than 1, then there exists $\epsilon_n>0$, such that the difference is at most $1 - \epsilon_n$. We also show that notion of $(k,d)$-colorings is equivalent to $r$-colorings (see (X. Zhu, Recent developments in circular coloring of graphs, in Topics in Discrete Mathematics Algorithms and Combinatorics Volume 26, Springer Berlin Heidelberg (2006) 497-550)).

Abstract: An acyclic homomorphism of a digraph $D$ into a digraph $F$ is a mapping $\phi\colon V(D) \to V(F)$ such that for every arc $uv\in E(D)$, either $\phi(u)=\phi(v)$ or $\phi(u)\phi(v)$ is an arc of $F$, and for every vertex $v\in V(F)$, the subgraph of $D$ induced on $\phi^{-1}(v)$ is acyclic. For each fixed digraph $F$ we consider the following decision problem: Does a given input digraph $D$ admit an acyclic homomorphism to $F$? We prove that this problem is NP-complete unless $F$ is acyclic, in which case it is polynomial time solvable. From this we conclude that it is NP-complete to decide if the circular chromatic number of a given digraph is at most $q$, for any rational number $q > 1$. We discuss the complexity of the problems restricted to planar graphs. We also refine the proof to deduce that certain $F$-coloring problems are NP-complete.