Circular Chromatic Number Research Articles

The circular chromatic number of signed series–parallel graphs of given girth

A signed graph is a graph G together with a signature σ:E(G)→{1,−1}. For a real number r≥1, Cr is a circle of circumference r. For two points a,b∈Cr, the distance d(modr)(a,b) between a and b is the length of the shorter arc of Cr connecting a and b. For x∈Cr, the antipodal x̄ of x is the unique point in Cr of distance r/2 from x. A circular r-colouring of (G,σ) is a mapping f:V(G)→Cr such that for each positive edge e=uv, d(modr)(f(u),f(v))≥1, and for each negative edge e=uv, d(modr)(f(u),f(v)¯)≥1. The circular chromatic number of a signed graph (G,σ) is the minimum r such that (G,σ) is circular r–colourable. Let g∗(G,σ) be the length of the shortest cycle in (G,σ) with an odd number of positive edges. For a positive integer g, let SPg be the family of signed series–parallel graphs with g∗(G,σ)≥g. This paper determines, for any positive integer g, the supremum value of χc(G,σ) of signed graphs in SPg.

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

A circular r-coloring of a signed graph (G,σ) is to assign points of a circle of circumference r, r≥2, to the vertices of G such that vertices connected by a positive edge are at circular distance at least 1 and vertices connected by a negative edge are at circular distance at most r2−1. The restriction of the notion of circular colorings to the class of signed bipartite graphs is already of high interest because the circular chromatic number of an (unsigned) graph can be obtained by bounding the circular chromatic number of an associated signed bipartite graph.In this paper, we define signed bipartite circular cliques Bp;qs and Bˆp;qs having the property that a signed bipartite graph admits a circular pq-coloring if and only if it admits an edge-sign preserving homomorphism to Bp;qs and a switching homomorphism to Bˆp;qs, respectively. Then as a bipartite analogue of Grötzsch's theorem, we prove that every signed bipartite planar graph of negative girth at least 6 admits a circular 3-coloring.

Circular \({\boldsymbol{(4-\epsilon )}}\) -Coloring of Some Classes of Signed Graphs

A circular -coloring of a signed graph is an assignment of points of a circle of circumference to the vertices of such that for each positive edge of the distance of from is at least 1 and for each negative edge the distance of from the antipode of is at least 1. The circular chromatic number of , denoted , is the infimum of such that admits a circular -coloring. This notion was recently defined by Naserasr, Wang, and Zhu, who, among other results, proved that for any signed -degenerate simple graph we have . For , examples of signed -degenerate simple graphs of circular chromatic number are provided. But for only examples of signed 2-degenerate simple graphs of circular chromatic number arbitrarily close 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 is a signed bipartite planar simple graph where vertices of one part are all of degree 2, then . Motivated by this observation, we provide an improved upper bound of for the circular chromatic number of a signed 2-degenerate simple graph on vertices and an improved upper bound of for the circular chromatic number of a signed bipartite planar simple graph on vertices. We then show that each of the bounds is tight for any value of .

Extremal decompositions for Nordhaus-Gaddum theorems

A Nordhaus-Gaddum theorem states bounds on p(G)+p(G‾) and p(G)⋅p(G‾) for some graph parameter p(G). We consider the sum upper bound for degeneracy, chromatic number, fractional and circular chromatic number, list chromatic number, span (for L(2,1) labeling), and point partition number. Viewing {G,G‾} as a decomposition of Kn, we describe a strategy to determine the extremal decompositions for these parameters. This produces short proofs of several existing results as well as several new theorems.

Describing hereditary properties by forbidden circular orderings

Each hereditary property can be characterized by its set of minimal obstructions; these sets are often unknown, or known but infinite. By allowing extra structure it is sometimes possible to describe such properties by a finite set of forbidden objects. This has been studied most intensely when the extra structure is a linear ordering of the vertex set. For instance, it is known that a graph G is k-colourable if and only if V(G) admits a linear ordering ≤ with no vertices v1≤⋯≤vk+1 such that vivi+1∈E(G) for every i∈{1,⋯,k}. In this paper, we study such characterizations when the extra structure is a circular ordering of the vertex set. We show that the classes that can be described by finitely many forbidden circularly ordered graphs include forests, circular-arc graphs, and graphs with circular chromatic number less than k. In fact, every description by finitely many forbidden circularly ordered graphs can be translated to a description by finitely many forbidden linearly ordered graphs. Nevertheless, our observations underscore the fact that in many cases the circular order descriptions are nicer and more natural.

Characterizing circular colouring mixing for pq<4 $\frac{p}{q}\lt 4$

AbstractGiven a graph , the ‐mixing problem asks: Starting with a ‐colouring of , can one obtain all ‐colourings of by changing the colour of only one vertex at a time, while at each step maintaining a ‐colouring? More generally, for a graph , the ‐mixing problem asks: Can one obtain all homomorphisms , starting from one homomorphism , by changing the image of only one vertex at a time, while at each step maintaining a homomorphism ? This paper focuses on a generalization of ‐colourings, namely, ‐circular colourings. We show that when , a graph is ‐mixing if and only if for any ‐colouring of , and any cycle of , the wind of the cycle under the colouring equals a particular value (which intuitively corresponds to having no wind). As a consequence we show that ‐mixing is closed under a restricted homomorphism called a fold. Using this, we deduce that ‐mixing is co‐NP‐complete for all , and by similar ideas we show that if the circular chromatic number of a connected graph is , then folds to . We use the characterization to settle a conjecture of Brewster and Noel, specifically that the circular mixing number of bipartite graphs is 2. Lastly, we give a polynomial time algorithm for ‐mixing in planar graphs when .

Signed planar graphs with given circular chromatic numbers

Disproving a conjecture of Máčajová, Raspaud and Škoviera, Kardoš and Narboni constructed a non-4-colourable simple signed planar graph, or equivalently, a simple signed planar graph with circular chromatic number greater than 4. Naserasr, Wang, and Zhu improved this result by constructing a simple signed planar graph with circular chromatic number 14/3. This paper further strengthens this result by constructing, for each rational 4<p/q≤14/3, a simple signed planar graph with circular chromatic number p/q. Together with some earlier results of Moser and Zhu, this implies that every rational p/q∈[2,14/3] is the circular chromatic number of a simple signed planar graph.

The circular chromatic numbers of signed series-parallel graphs

It is known that any signed series-parallel graph (G,σ) has circular chromatic number at most 10/3. This paper proves that for each rational r∈[2,10/3], there is a signed series-parallel graph (G,σ) whose circular chromatic number equals r.

Circular Chromatic Number of Signed Graphs

A signed graph is a pair $(G, \sigma)$, where $G$ is a graph (loops and multi edges allowed) and $\sigma: E(G) \to \{+, -\}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend circular coloring of graphs to signed graphs. Given a signed graph $(G, \sigma)$ with no positive loop, a circular $r$-coloring of $(G, \sigma)$ is an assignment $\psi$ of points of a circle of circumference $r$ to the vertices of $G$ such that for every edge $e=uv$ of $G$, if $\sigma(e)=+$, then $\psi(u)$ and $\psi(v)$ have distance at least $1$, and if $\sigma(e)=-$, then $\psi(v)$ and the antipodal of $\psi(u)$ have distance at least $1$. The circular chromatic number $\chi_c(G, \sigma)$ of a signed graph $(G, \sigma)$ is the infimum of those $r$ for which $(G, \sigma)$ admits a circular $r$-coloring. For a graph $G$, we define the signed circular chromatic number of $G$ to be $\max\{\chi_c(G, \sigma): \sigma \text{ is a signature of $G$}\}$. &#x0D; We study basic properties of circular coloring of signed graphs and develop tools for calculating $\chi_c(G, \sigma)$. We explore the relation between the circular chromatic number and the signed circular chromatic number of graphs, and present bounds for the signed circular chromatic number of some families of graphs. In particular, we determine the supremum of the signed circular chromatic number of $k$-chromatic graphs of large girth, of simple bipartite planar graphs, $d$-degenerate graphs, simple outerplanar graphs and series-parallel graphs. We construct a signed planar simple graph whose circular chromatic number is $4+\frac{2}{3}$. This is based and improves on a signed graph built by Kardos and Narboni as a counterexample to a conjecture of Máčajová, Raspaud, and Škoviera.

Coloring graphs by translates in the circle

The fractional and circular chromatic numbers are the two most studied non-integral refinements of the chromatic number of a graph. Starting from the definition of a coloring base of a graph, which originated in work related to ergodic theory, we formalize the notion of a gyrocoloring of a graph: the vertices are colored by translates of a single Borel set in the circle group, and neighboring vertices receive disjoint translates. The corresponding gyrochromatic number of a graph always lies between the fractional chromatic number and the circular chromatic number. We investigate basic properties of gyrocolorings. In particular, we construct examples of graphs whose gyrochromatic number is strictly between the fractional chromatic number and the circular chromatic number. We also establish several equivalent definitions of the gyrochromatic number, including a version involving all finite abelian groups.

T-colorings, divisibility and circular chromatic number

Let T be a T-set, i.e., a finite set of nonnegative integers satisfying 0 ∈ T, and G be a graph. In the paper we study relations between the T-edge spans espT (G) and espd⊙T (G), where d is a positive integer and d ⊙ T = {0 ≤ t ≤ d (max T + 1): d |t ⇒ t/d ∈ T} . We show that espd⊙T (G) = d espT (G) − r, where r, 0 ≤ r ≤ d − 1, is an integer that depends on T and G. Next we focus on the case T = {0} and show that espd⊙{0} (G) = ⌈d(χc(G) − 1)⌉, where χc(G) is the circular chromatic number of G. This result allows us to formulate several interesting conclusions that include a new formula for the circular chromatic number χc(G) = 1 + inf espd⊙{0} (G)/d: d ≥ 1 and a proof that the formula for the T-edge span of powers of cycles, stated as conjecture in [Y. Zhao, W. He and R. Cao, The edge span of T-coloring on graph C d n , Appl. Math. Lett. 19 (2006) 647–651], is true.

On the Circular Altitude of Graphs

Peter Cameron introduced the concept of the circular altitude of graphs, a parameter which was shown by Bamberg et al. that provides a lower bound on the circular chromatic number. In this paper, we investigate this parameter and show that the circular altitude of a graph is equal to the maximum of circular altitudes of its blocks. Also, we show that homomorphically equivalent graphs have the same circular altitudes. Finally, we prove that the circular altitude of the Cartesian product of two graphs is equal to the maximum of circular altitudes of its factors.

Maximal Density of Sets with Missing Differences and Various Coloring Parameters of Distance Graphs

For a given set $M$ of positive integers, a well-known problem of Motzkin asked to determine the maximal asymptotic density of $M$-sets, denoted by $\mu(M)$, where an $M$-set is a set of non-negative integers in which no two elements differ by an element in $M$. In 1973, Cantor and Gordon found $\mu(M)$ for $|M| \leq 2$. Partial results are known in the case $|M| \geq 3$ including results in the case when $M$ is an infinite set. This number theory problem is also related to various types of coloring problems of the distance graphs generated by $M$. In particular, it is known that the reciprocal of the fractional chromatic number of the distance graph generated by $M$ is equal to the value $\mu(M)$ when $M$ is finite. Motivated by the families $M = \{ a,b,a+b \}$ and $M = \{ a,b,a+b,b-a \}$ discussed by Liu and Zhu, we study two families of sets $M$, namely, $M = \{ a,b,b-a,n(a+b) \}$ and $M = \{ a,b,a+b,n(b-a) \}$. For both of these families, we find some exact values and some bounds on $\mu(M)$. We also find bounds on the fractional and circular chromatic numbers of the distance graphs generated by these families. Furthermore, we determine the exact values of chromatic number of the distance graphs generated by these two families.

A topological proof of Chen's alternative Kneser coloring theorem

Johnson, Holroyd and Stahl [5] conjectured that the circular chromatic number of the Kneser graph is equal to the ordinary chromatic number. Chen completely confirmed the conjecture in [4]. Chen's alternative Kneser coloring theorem is a key lemma in his proof of Johnson-Holroyd-Stahl conjecture. Chen [4] and Chang, Liu and Zhu [3] proved the theorem by using Fan's lemma. In this paper, we prove Chen's alternative Kneser coloring theorem by using cohomology.

On the chromatic number of a subgraph of the Kneser graph

Let n and k be positive integers with n≥2k. Consider a circle C with n points 1,…,n in clockwise order. The interlacing graphIGn,k is the graph with vertices corresponding to k-subsets of [n] that do not contain two adjacent points on C, and edges between k-subsets P and Q if they interlace: after removing the points in P from C, the points in Q are in different connected components. In this paper we prove that the circular chromatic number of IGn,k is equal to n/k, hence the chromatic number is ⌈n/k⌉, and that its independence number is (n−k−1k−1).

Circular chromatic number of induced subgraphs of Kneser graphs

Investigating the equality of the chromatic number and the circular chromatic number of graphs has been an active stream of research for last decades. In this regard, Hajiabolhassan and Zhu in 2003 proved that if n is sufficiently large with respect to k , then the Schrijver graph SG( n , k ) has the same chromatic and circular chromatic number. Later, Meunier in 2005 and independently, Simonyi and Tardos in 2006 proved that χ (SG( n , k )) = χ c (SG( n , k )) if n is even. In this paper, we study the circular chromatic number of induced subgraphs of Kneser graphs. In this regard, we shall first generalize the preceding result to s -stable Kneser graphs for large even n and even s . Furthermore, as a generalization of the Hajiabolhassan-Zhu result, we prove that if n is large enough with respect to k , then any sufficiently large induced subgraph of the Kneser graph KG( n , k ) has the same chromatic number and circular chromatic number.

On Chromatic Numbers of Two Extensions of Planar Graphs

In this paper, the acyclic chromatic and the circular list chromatic numbers of a simple H-minor free graph G is considered, where H ∈{K5,K3,3}. It is proved that the acyclic chromatic number (resp. the circular list chromatic number) of a simple H-minor free graph G where H ∈{K5,K3,3} is at most 5 (resp. at most 8) and we conclude that G is star 20-colorable. These results generalize the same known results on planar graphs. Moreover, some upper bounds for the coloring numbers of H-minor free graphs for H ∈{K5,K3,3,Kr,s} and r ≤ 2 are obtained. These results generalize some known results and give some new results on group choice number, group chromatic number, and the choice number of the mentioned graphs with much shorter proofs.

Upper Bound on the Circular Chromatic Number of the Plane

We consider circular version of the famous Nelson-Hadwiger problem. It is know that 4 colors are necessary and 7 colors suffice to color the euclidean plane in such a way that points at distance one get different colors. In $r$-circular coloring we assign arcs of length one of a circle with a perimeter $r$ in such a way that points at distance one get disjoint arcs. In this paper we show the existence of $r$-circular coloring for $r=4+\frac{4\sqrt{3}}{3}\approx 6.30$. It is the first result with $r$-circular coloring of the plane with $r$ smaller than 7. We also show $r$-circular coloring of the plane with $r&lt;7$ in the case when we require disjoint arcs for points at distance belonging to the internal [0.9327,1.0673].

The Vietoris–Rips complexes of a circle

Given a metric space X and a distance threshold r>0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Little is known about the behavior of Vietoris-Rips complexes for larger values of r, even though these complexes arise naturally in applications using persistent homology. We show that as r increases, the Vietoris-Rips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. As our main tool we introduce a directed graph invariant, the winding fraction, which in some sense is dual to the circular chromatic number. Using the winding fraction we classify the homotopy types of the Vietoris-Rips complex of an arbitrary (possibly infinite) subset of the circle, and we study the expected homotopy type of the Vietoris-Rips complex of a uniformly random sample from the circle. Moreover, we show that as the distance parameter increases, the ambient Cech complex of the circle also obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible.

Strengthening topological colorful results for graphs

Various results ensure the existence of large complete and colorful bipartite graphs in properly colored graphs when some condition related to a topological lower bound on the chromatic number is satisfied. We generalize three theorems of this kind, respectively due to Simonyi and Tardos 2006), Simonyi et al. (2013), and Chen 2011). As a consequence of the generalization of Chen’s theorem, we get new families of graphs whose chromatic number equals their circular chromatic number and that satisfy Hedetniemi’s conjecture for the circular chromatic number.