PREPRINT
國立臺灣大學 數學系 預印本 Department of Mathematics, National Taiwan University
www.math.ntu.edu.tw/ ~ mathlib/preprint/2012- 05.pdf
A Short Proof for Chen’s Alternative Kneser Coloring Lemma
Gerard Jennhwa Chang, Daphne Der-Fen Liu, and Xuding Zhu
April 16, 2012
Gerard Jennhwa Chang∗ Daphne Der-Fen Liu† Xuding Zhu‡ April 16, 2012
Abstract
We give a short proof for Chen’s Alternative Kneser Coloring Lemma. This leads to a short proof for Johnson-Holroyd-Stahl’s conjecture that Kneser graphs have their circular chromatic numbers equal to their chromatic numbers.
1 Introduction
Suppose G is a graph and p ≥ q ≥ 1 are integers. A (p, q)-coloring of G is a mapping c: V (G)→ {0, 1, . . . , p − 1} such that q ≤ |f(x) − f(y)| ≤ p − q for every edge xy of G. A graph is (p, q)-colorable if it admits a (p, q)-coloring. The circular chromatic number of G is
χc(G) = inf{p/q: G is (p, q)-colorable}.
It is well-known [8] that for any graph G, χ(G)− 1 < χc(G)≤ χ(G). The question as which graphs G satisfy the equality χc(G) = χ(G) has received considerable attention.
Given positive integers n ≥ 2k, the Kneser graph KG(n, k) has vertex set ([n]k), i.e., all k-subsets of [n] = {1, 2, . . . , n}, in which two vertices A and B are adjacent if A ∩ B = ∅.
Coloring of Kneser graphs has been a fascinating subject in graph theory. In proving Kneser’s conjecture that χ(KG(n, k)) = n− 2k + 2, Lov´asz [5] initiated the application of algebraic topology to graph coloring. Since then, this method has became an important tool with wide applications in combinatorics.
Johnson, Holroyd and Stahl [4] first studied the circular chromatic number of Kneser graphs, and conjectured that the equality χc(KG(n, k)) = χ(KG(n, k)) always holds. This conjecture has received a lot of attention. Hajiabolhassan and Zhu [3] proved that for a fixed k, if n is sufficiently large, then χc(KG(n, k)) = χ(KG(n, k)). Meunier [6] and Simonyi and Tardos [7]
proved independently that if n is even then χc(KG(n, k)) = χ(KG(n, k)). The proof in [3] is combinatorial, and the proofs in [6, 7] use Fan’s Lemma from algebraic topology. Nevertheless, both proofs also apply to Schrijver graphs SG(n, k) (subgraphs of KG(n, K) induced by stable k-subsets as vertices). On the other hand, it is known [7] that if n is odd and is not much
∗Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan. Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan. National Center for Theoretical Sciences, Taipei Office, Taiwan. E-mail:[email protected]. Supported in part by the National Science Council under grant NSC98-2115-M-002-013-MY3.
†Department of Mathematics, California State University, Los Angeles, U.S.A. E-mail: [email protected].
‡Department of Mathematics, Zhejiang Normal University, China. Grant Number: NSF11171310 and ZJNSF Z6110786. E-mail: [email protected].
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bigger than 2k, then χc(SG(n, k)) ̸= χ(SG(n, k)). So it seemed not of much hope to apply these methods to completely prove Johnson-Holroyd-Stahl’s conjecture.
However, recently Chen [1] completely proved Johnson-Holroyd-Stahl’s conjecture by using Fan’s Lemma in an innovative way. A key step in Chen’s proof is to prove the Alternative Kneser Coloring Lemma. Assume Kq,q is a complete bipartite graph with partite sets X = {x1, x2, . . . , xq} and Y = {y1, y2, . . . , yq}. Denote by Kq,q∗ the graph obtained from Kq,q by deleting the edges of a perfect matching, say by deleting the edges xiyi (i = 1, 2, . . . , q).
Assume Kq,q∗ is a subgraph G and c is a q-coloring of G. We say Kq,q∗ is colorful with respect to c of if c(xi) = c(yi). Observe that if Kq,q∗ is colorful with respect to a q-coloring c, then c(xi)̸= c(xj) for i̸= j, and hence we may assume that c(xi) = c(yi) = i for i = 1, 2, . . . , q.
Lemma 1 (Alternative Kneser Coloring Lemma [1]) Any proper (n− 2k + 2)-coloring of KG(n, k) contains a colorful copy of Kn∗−2k+2,n−2k+2.
Note that Lov´asz’s result is equivalent to say that for every (n−2k+2)-coloring of KG(n, k), each color class is non-empty. Chen’s Alternative Kneser Coloring Lemma reveals a more deli- cate structure of (n−2k+2)-colorings for KG(n, k). Besides its application to the determination of the circular chromatic number of Kneser graphs, the lemma is interesting by itself.
Chen’s proof of Lemma 1 is rather complicated. In this article, we give a shorter proof for this result. Before presenting it, for completeness, we show how Lemma 1 is used to settle the Johnson-Holroyd-Stahl’s conjecture.
Lemma 2 If G is q-colorable and every q-coloring of G contains a colorful copy of Kq,q∗ , then χc(G) = χ(G) = q.
Proof. For a q-coloring c of G, a cycle C = (v0, v1, . . . , vn−1, v0) is called tight if c(vi+1) ≡ c(vi) + 1 (mod q) for i = 0, 1, . . . , n− 1, where the indices of the vertices are modulo n. It is known [8] that χc(G) = q if and only if G is q-colorable and every q-coloring of G has a tight cycle. The assumption of Lemma 2 implies that every q-coloring c of G has a tight cycle.
Assume a colorful copy of Kq,q∗ with respect to c has partite sets X = {x1, x2, . . . , xq} and Y = {y1, y2, . . . , yq}, with c(xi) = c(yi) = i for i = 1, 2, . . . , q. If q is even, then (x1, y2, x3, y4, . . . , xq−1, yq, x1) is a tight cycle. If q is odd, then (x1, y2, x3, y4, . . ., yq−1, xq, y1, x2, y3, x4, . . ., xq−1, yq, x1) is a tight cycle. Thus, χc(G) = q.
Johnson-Holroyd-Stahl’s conjecture is an immediate consequence of Lemmas 1 and 2.
2 Proof of Alternative Kneser Coloring Lemma
We use Fan’s Lemma to prove Chen’s Alternative Kneser Coloring Lemma. Let n be a positive integer and let [−1, 1]n = {x ∈ Rn:||x||∞ ≤ 1} be the n-dimensional cube. The barycentric subdivision of [−1, 1]n, denoted by sd([−1, 1]n), is the simplicial complex whose vertices are points in [−1, 1]n with each coordinate either 0, 1 or −1, and a set of vertices form a simplex if the vertices can be ordered as v1, v2, . . . , vt so that for i = 1, 2, . . . , t− 1, if a coordinate of vi is 1 (or −1, respectively) then the corresponding coordinate of vi+1 is also 1 (or −1, respectively). The simplicial complex sd([−1, 1]n) is a triangulation of [−1, 1]n. The boundary of sd([−1, 1]n), denoted by ∂(sd([−1, 1]n)), is a triangulation of the (n− 1)-dimensional sphere
Sn−1. Each vertex in ∂(sd([−1, 1]n)) is a vector in{−1, 1, 0}n\ {0}n. We denote such a vector by a signed set X, which is a pair X = (X+, X−) of disjoint subsets X+, X− ⊆ [n], defined as X+ ={i: X(i) = 1} and X− ={i: X(i) = −1}. Let |X| = |X+| + |X−|, and write X ≤ Y if X+ ⊆ Y+ and X− ⊆ Y−, and write X < Y if X ≤ Y and X ̸= Y . Thus a simplex in
∂(sd([−1, 1]n)) is a sequence of signed sets ∅ ̸= X1 < X2< . . . < Xt.
An n-labeling of ∂(sd([−1, 1]n)) is a mapping λ:{−1, 1, 0}n\{0}n→ {±1, ±2, . . . , ±n}. An n-labeling λ is antipodal if λ(−X) = −λ(X) for all X. A complementary edge with respect to λ is a pair of signed sets X, Y such that X < Y and λ(X) =−λ(Y ). A simplex X1< X2 < . . . <
Xn is a positive alternating (n− 1)-simplex with respect to λ if {λ(X1), λ(X2), . . . , λ(Xn)} = {1, −2, . . . , (−1)n−1n}. The following is a special case of Fan’s Lemma.
Octahedral Fan’s Lemma [2] If λ is an antipodal n-labeling of the vertices of ∂(sd([−1, 1]n)) without complementary edges, then the number of positive alternating (n− 1)-simplices is odd.
To apply Fan’s Lemma, we shall associate to each proper (n− 2k + 2)-coloring of KG(n, k) with a labeling for the vertices of ∂(sd([−1, 1]n)). Chen’s proof of the Alternative Kneser Coloring Lemma also uses this approach. The difference between the two proofs is the labelings associated to the colorings of KG(n, k). Chen’s labeling is the composition of two functions, including a rather complicated one, while the labeling we use is direct and simple.
Assume c is a proper (n− 2k + 2)-coloring of KG(n, k), using colors from the set {2k − 1, 2k, . . . , n}. For a subset S of [n] with |S| ≥ k, let
c(S) = max{c(A): A ⊆ S, |A| = k}.
Let ≺ be an arbitrary linear ordering on subsets of [n] such that X ≺ Y implies |X| ≤ |Y |.
Let λ:{−1, 1, 0}n\ {0}n→ {±1, ±2, . . . , ±n} be defined as follows:
λ(X) =
|X|, if|X| ≤ 2k − 2 and X−≺ X+;
−|X|, if|X| ≤ 2k − 2 and X+≺ X−; c(X+), if|X| ≥ 2k − 1 and X−≺ X+;
−c(X−), if|X| ≥ 2k − 1 and X+≺ X−.
It is obvious that λ is antipodal and it is easy to verify that there is no complementary edge.
By Fan’s Lemma, there are an odd number of positive alternating (n− 1)-simplices.
Assume X1 < X2 < . . . < Xn is a positive alternating (n− 1)-simplex with respect to λ.
Since 1≤ |X1| < |X2| < . . . < |Xn| ≤ n, one has |Xi| = i for 1 ≤ i ≤ n.
Claim 1. Let X0 = (∅, ∅). For any index 1 ≤ i ≤ n, either |Xi+| = |Xi+−1| + 1, Xi−−1 = Xi− ≺ Xi+ and λ(Xi) > 0, or else |Xi−| = |Xi−−1| + 1, Xi+−1 = Xi+≺ Xi− and λ(Xi) < 0.
Proof. For 1 ≤ i ≤ 2k − 2, it follows from the definitions of λ and the positive alternating (n− 1)-simplices that λ(Xi) = (−1)i−1i, and hence if i is odd, then |Xi+| = |Xi+−1| + 1 and Xi−−1 = Xi− ≺ Xi+; if i is even, then |Xi−| = |Xi−−1| + 1, Xi+−1 = Xi+ ≺ Xi−. In particular,
|X2k+−2| = |X2k−−2| = k − 1.
Assume 2k− 1 ≤ i ≤ n. Since Xi−1 < Xi and |Xi| = |Xi−1| + 1, we know that either
|Xi+| = |Xi+−1| + 1 and Xi−−1 = Xi− or else |Xi−| = |Xi−−1| + 1 and Xi+−1 = Xi+. Assume
|Xi+| = |Xi−1+ | + 1 (the case |Xi−| = |Xi−1− | + 1 is symmetric). Assume to the contrary of the claim that Xi+ ≺ Xi−. Then |Xi+| ≤ |Xi−| and so |Xi+−1| < |Xi−−1| which gives Xi+−1 ≺ Xi−−1
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and i− 1 ̸= 2k − 2. Hence, λ(Xi) =−c(Xi−) = −c(Xi−−1) = λ(Xi−1), contradicting the fact that λ(Xr)̸= λ(Xs) for r̸= s. 2
Since ⌈n/2⌉ of the labels λ(Xi)’s are positive and ⌊n/2⌋ of them are negative, it follows from Claim 1 that |Xn+| = ⌈n/2⌉ and |Xn−| = ⌊n/2⌋.
Claim 2. For any index 1≤ i ≤ n, it holds that −1 ≤ |Xi+| − |Xi−| ≤ 1.
Proof. By symmetry, it is enough to show that |Xi+| − |Xi−| ≤ 1. Assume to the contrary that |Xi+| − |Xi−| ≥ 2 for some i. Since |Xn+| − |Xn−| ≤ 1, there is an index j such that
|Xj+1+ | − |Xj+1− | ≤ 1 < 2 ≤ |Xj+| − |Xj−|. Hence |Xj+1− | = |Xj−| + 1. By Claim 1, Xj+1+ ≺ Xj+1− and so |Xj+1+ | ≤ |Xj+1− |, which is impossible as |Xj+| − |Xj−| ≥ 2. 2
It follows from Claim 2 that |X2j+| = |X2j−| = j for 1 ≤ j ≤ n/2. So we may denote [n] ={a1, a2, . . . , an} where X2j+ ={a1, a3, . . . , a2j−1} and X2j− ={a2, a4, . . . , a2j}. The signed set X2j−1 can be either (X2j+, X2j−−2) or (X2j+−2, X2j−).
As observed above, λ(Xi) = (−1)i−1i for 1 ≤ i ≤ 2k − 2. For 2k − 1 ≤ i ≤ n, since {λ(X2k−1), λ(X2k), . . . , λ(Xn)} = {2k − 1, −2k, . . . , (−1)n−1n}, by the monotonicity of c,
c({a1, a3, . . . , ai}) = i for odd i; and c({a2, a4, . . . , ai}) = i for even i.
Let Γ ={X ∈ {+, −, 0}n:|X+| = |X−| = k − 1}. As noted above, each positive alternating (n− 1)-simplex contains exactly one vertex in Γ. For X ∈ Γ, let α(X, λ) be the number of positive alternating (n− 1)-simplices containing vertex X. By Fan’s Lemma, ΣX∈Γα(X, λ) is odd. Hence there exists Z ∈ Γ such that α(Z, λ) is odd. In particular, there exists a positive alternating (n− 1)-simplex X1 < X2< . . . < Xn with respect to λ, with Z = X2k−2. For this Z, define λ′:{+, −, 0}n\ {0}n→ {±1, ±2, . . . , ±n} by:
λ′(X) =
{ −λ(X), if X ∈ {Z, −Z};
λ(X), otherwise.
Then λ′is antipodal without complementary edges. By Fan’s Lemma, there are an odd number of positive alternating (n − 1)-simplices with respect to λ′. Since α(X, λ′) = α(X, λ) for X∈ Γ \ {Z, −Z}, we conclude that
α(Z, λ) + α(−Z, λ) ≡ α(Z, λ′) + α(−Z, λ′) (mod 2).
Since λ(Z) =−(2k−2) and so λ(−Z) = 2k−2 = λ′(Z), we know that α(−Z, λ) = α(Z, λ′) = 0.
Thus, α(−Z, λ′)≡ α(Z, λ) ≡ 1 (mod 2). So there exists a positive alternating (n − 1)-simplex Y1 < Y2 < . . . < Yn with respect to λ′, where Y2k−2 =−Z. Similar to the discussion for λ, we may denote [n] = {b1, b2, . . . , bn} where Y2j+ = {b1, b3, . . . , b2j−1} and Y2j− = {b2, b4, . . . , b2j}.
The signed set Y2j−1 can be either (Y2j+, Y2j−−2) or (Y2j+−2, Y2j−), where Y0+= Y0− =∅. Also, for 2k− 1 ≤ i ≤ n, c({b1, b3, . . . , bi}) = i for odd i; and c({b2, b4, . . . , bi}) = i for even i.
Let Z = (S, T ). Then X2k−2= (S, T ) and Y2k−2= (T, S). Consequently, for 2k−1 ≤ i ≤ n, c(S∪ {a2k−1, a2k+1, . . . , ai}) = c(T ∪ {b2k−1, b2k+1, . . . , bi}) = i for odd i; and
c(T ∪ {a2k, a2k+2, . . . , ai}) = c(S ∪ {b2k, b2k+2, . . . , bi}) = i for even i.
Claim 3. For any index 2k−1 ≤ i ≤ n, it holds that ai = bi and c(S∪{ai}) = c(T ∪{ai}) = i.
Proof. We prove by induction on i. If i = 2k−1, since c(S ∪{a2k−1}) = c(T ∪{b2k−1}) = 2k−1, one has that S∪ {a2k−1} and T ∪ {b2k−1} are not adjacent, implying a2k−1 = b2k−1. Assume i ≥ 2k and the claim is true for i′ < i. Since for all 2k− 1 ≤ j < i, the vertex S ∪ {ai} is adjacent to T ∪ {aj}, we conclude that c(S ∪ {ai}) ̸= c(T ∪ {aj}) = j for 2k − 1 ≤ j < i.
Therefore c(S∪ {ai}) = i. Similarly, c(T ∪ {bi}) = i. As c(S ∪ {ai}) = c(T ∪ {bi}), one has that S∪ {ai} and T ∪ {bi} are not adjacent. Hence ai = bi. This completes the proof of the claim. 2
The subgraph of KG(n, k) induced by the vertices {S ∪ {ai}, T ∪ {ai}: 2k − 1 ≤ i ≤ n}
is a colorful copy of Kn∗−2k+2,n−2k+2. This completes the proof of Chen’s Alternative Kneser Coloring Lemma.
References
[1] P.-A. Chen. A new coloring theorem of Kneser graphs. J. Combin. Theory Ser. A, 118(3):1062–1071, 2011.
[2] K. Fan. A generalization of Tucker’s combinatorial lemma with topological applications.
Ann. of Math. (2), 56:431–437, 1952.
[3] H. Hajiabolhassan and X. Zhu. Circular chromatic number of Kneser graphs. J. Combin.
Theory Ser. B, 88(2):299–303, 2003.
[4] A. Johnson, F. C. Holroyd and S. Stahl. Multichromatic numbers, star chromatic numbers and Kneser graphs. J. Graph Theory, 26(3):137–145, 1997.
[5] L. Lov´asz. Kneser’s conjecture, chromatic number, and homotopy. J. Combin. Theory Ser. A, 25(3):319–324, 1978.
[6] F. Meunier. A topological lower bound for the circular chromatic number of Schrijver graphs. J. Graph Theory, 49(4):257–261, 2005.
[7] G. Simonyi and G. Tardos. Local chromatic number, Ky Fan’s theorem and circular colorings. Combinatorica, 26(5):587–626, 2006.
[8] X. Zhu. Circular chromatic number: a survey. Discrete Math., 229(1-3):371–410, 2001.
