CIRCULAR COLOURING THE PLANE MATT DEVOS, JAVAD EBRAHIMI, MOHAMMAD GHEBLEH, LUIS GODDYN, BOJAN MOHAR, AND REZA NASERASR Astrct. The unit distnce grph R is the grph with vertex set R 2 in which two vertices (points in the plne) re djcent if nd only if they re t Eucliden distnce 1. We prove tht the circulr chromtic numer ofris t lest 4, thus improving the known lower ound of 32/9 otined from the frctionl chromtic numer ofr. Key words. grph colouring, circulr colouring, unit distnce grph AMS suject clssifictions. 05C15, 05C10, 05C62 1. Introduction. The unit distnce grph R is defined to e the grph with vertex set R 2 in which two vertices (points in the plne) re djcent if nd only if they re t Eucliden distnce 1. Every sugrph of R is lso sid to e unit distnce grph. It is known tht (cf. [1, 2]) nd tht (cf. [3, pp. 59 65]) 4 χ(r) 7, 32 9 χ f(r) 4.36. Here χ(r) denotes the chromtic numer of R, nd χ f (R) is the frctionl chromtic numer of R defined s follows: fold colouring of grph G is n ssignment of sets of colours to the vertices of G. The frctionl chromtic numer of G, denoted χ f (G), is defined y χ f (G) = inf{ G hs fold colouring using colours}. In this pper we study the circulr chromtic numer of the unit distnce grph R. Let r 2,, [0, r), nd. We define the circulr distnce of nd, denoted y δ(, ) = δ r (, ), to e min{, r+ }. One my identify the intervl [0, r) with circle C r with perimeter r nd then δ(, ) will e the distnce etween nd in C r. It is esy to see tht δ stisfies the tringle inequlity. If, [0, r) (or equivlently, C r ), we define the circulr intervl from to, denoted [, ], s follows (see Figure 1.1): { {x x } if, [, ] = {x 0 x or x < r} if >. An r-circulr colouring of grph G, is function c : V (G) C r such tht for every edge xy in G, δ(c(x), c(y)) 1. The circulr chromtic numer of G, denoted y χ c (G), is χ c (G) = inf{r G dmits n r-circulr colouring}. Deprtment of Mthemtics, Simon Frser University, Burny, British Columi, V5A 1S6, Cnd. Deprtment of Comintorics nd Optimiztion, Fculty of Mthemtics, University of Wterloo, Wterloo, Ontrio, N2L 3G1, Cnd. 1
2 DEVOS, EBRAHIMI, GHEBLEH, GODDYN, MOHAR, NASERASR 0 0 < > Fig. 1.1. Circulr intervls (clockwise direction is the positive direction) x y Fig. 2.1. The unit distnce grph H, It is well known [4] tht for every grph G, χ f (G) χ c (G) χ(g). For the unit distnce grph R, these inequlities give 32 9 χ f(r) χ c (R) χ(r) 7. We improve the lower ound for χ c (R) to 4. We give two proofs of this result. The second one is constructive nd gives construction of finite unit distnce grphs with circulr chromtic numer ritrrily close to 4. 2. Proof. Let nd e two points in the plne nd let d(, ) denote the Eucliden distnce etween nd. If d(, ) = 3, then we my find points x nd y in the plne such tht the sugrph of R induced on the set {,, x, y} is isomorphic to the grph H otined y deleting one edge from K 4 (see Figure 2.1). We denote this unit distnce grph y H,. On the other hnd, it is esy to see tht in ny emedding of H s unit distnce grph in the plne, the Eucliden distnce etween the two vertices of degree 2 in H is 3. Lemm 2.1. Let 0 < ε < 1 nd, R 2 with d(, ) = 3. Let c e (3 + ε)- circulr colouring of H,. Then δ(c(), c()) ε. Proof. Without loss of generlity, we my ssume c() = 0. Since, x, y form tringle in H,, we hve c(x) [1, 1 + ε] nd c(y) [2, 2 + ε] up to symmetry. On the other hnd, is djcent to oth x nd y. Thus c() [c(x) + 1, c(x) 1] [c(y) + 1, c(y) 1] [2, ε] [ ε, 1 + ε] = [ ε, ε]. The lst equlity is true since 1 + ε < 2. Theorem 2.2. χ c (R) 4.
CIRCULAR COLOURING THE PLANE 3 Proof. Suppose tht c is (3 + ε)-circulr colouring of R where 0 ε < 1. Let µ = sup{δ(c(), c()), R 2 nd d(, ) = 3}. By Lemm 2.1, µ ε. By the definition of µ, for every 0 < µ < µ, there exist points nd t distnce 3 in the plne such tht δ(c(), c()) > µ. Consider the grph H, s in Figure 2.1. Without loss of generlity we my ssume Since 3 + ε < 4, we hve 0 = c() c() < c(x) < c(y) 2 + ε. δ(c(), c(x)) = c(x) = δ(c(), c()) + δ(c(), c(x)) > µ + 1. On the other hnd since nd x re t distnce 1, there exists point z which is t distnce 3 from oth nd x. Therefore 1 + µ < δ(c(), c(x)) δ(c(), c(z)) + δ(c(z), c(x)) 2µ. Since this is true for every µ < µ, we hve µ 1. This is contrdiction since µ ε < 1. 3. A constructive proof. The grph G 0 = K 2 is oviously unit distnce grph. In our construction of grphs (n 0) we distinguish two vertices in ech of them. To emphsize the distinguished vertices x nd y of, we write G x,y n. We identify sugrphs of R with their geometric representtion given y their vertex set. For n 0, the grph +1 is constructed recursively from four copies of. Let S = V (G x,y n ) R 2. Let us rotte the set S in the plne out the point x, so tht the imge y of y under this rottion is t distnce 1 from y. Let S e the imge of S under this rottion. Let T e the set of ll points in S S nd their reflections cross the line yy. In prticulr let z T e the reflection of x cross the line yy. We define G x,z n+1 to e the sugrph of R induced on T. This construction is depicted in Figure 3.1. y x z y Fig. 3.1. Construction of +1 from Note tht G 1 is the grph H, of Figure 2.1 nd G 2 contins the Moser grph shown in Figure 3 s sugrph. The Moser grph, lso known s the spindle grph, ws the first 4 chromtic unit distnce grph discovered [2]. Lemm 3.1. For every n 1, χ c ( ) 4 2 1 n. Moreover, for every r = 4 2 1 n + ε with 0 ε < 2 1 n, nd every circulr r-colouring c of G x,z n, we hve δ(c(x), c(z)) 2 n 1 ε. Proof. We use induction on n. The nontrivil prt of the cse n = 1 is proved in Lemm 2.1. Let n 1 nd G x,z n+1 e s shown in Figure 3.1. Let r = 4 21 n + ε
4 DEVOS, EBRAHIMI, GHEBLEH, GODDYN, MOHAR, NASERASR Fig. 3.2. The Moser (spindle) grph for some ε 0 nd let c e circulr r-colouring of G x,z n+1. Without loss of generlity we my ssume tht c(x) = 0. By the induction hypothesis, δ(0, c(y)) nd δ(0, c(y )) re oth t most 2 n 1 ε. Hence δ(c(y), c(y )) 2 n ε. On the other hnd, since y nd y re djcent in G x,z n+1, we hve δ(c(y)), c(y )) 1. Therefore ε 2 n nd we hve χ c (+1 ) 4 2 1 n + 2 n = 4 2 n. Now let r = 4 2 n + ε for some 0 ε < 2 n, nd let c e circulr r-colouring of +1 with c(x) = 0. Note tht r = 4 2 1 n + ε with ε = 2 n + ε < 2 1 n. By the induction hypothesis, δ(0, c(y)), δ(0, c(y )), δ(c(z), c(y)) nd δ(c(z), c(y )) re ll t most 2 n 1 ε < 1. Therefore we hve nd c(y), c(y ) [ 2 n 1 ε, 2 n 1 ε ] c(z) [c(y) 2 n 1 ε, c(y) + 2 n 1 ε ] [c(y ) 2 n 1 ε, c(y ) + 2 n 1 ε ]. Since δ(c(y), c(y )) 1, one of c(y) nd c(y ), sy c(y), is in the circulr intervl [ 2 n 1 ε, 2 n 1 ε 1], nd c(y ) [ 2 n 1 ε + 1, 2 n 1 ε ]. Therefore nd [c(y) 2 n 1 ε, c(y) + 2 n 1 ε ] [ 2 n ε, 2 n ε 1] = [ 2 n ε, 2 n ε] [c(y ) 2 n 1 ε, c(y ) + 2 n 1 ε ] [ 2 n ε + 1, 2 n ε ] = [ 2 n ε, 2 n ε ]. Finlly, since ε < 2 1 n, we hve 2 n ε < r 2 n ε. Hence c(z) [ 2 n ε, 2 n ε] [ 2 n ε, 2 n ε ] = [ 2 n ε, 2 n ε]. This completes the induction step. Let us oserve tht, when constructing +1 from four copies of, it my hppen tht vertices in distinct copies of correspond to the sme points in the plne. Additionlly, it my hppen tht some edges etween vertices in distinct copies of re introduced. We my define in the sme wy sequence of strct grphs H n, where none of these two issues occur. Clerly χ c ( ) χ c (H n ), ut we cnnot rgue equlity in generl. The proof of Lemm 3.1 pplied to the grphs H n gives slightly more: Theorem 3.2. For every n 0, χ c (H n ) = 4 2 1 n. Proof. The cses n = 0, 1 re trivil. Let n 1 nd let H n+1 e s in Figure 3.1. Let r = 4 2 n = 4 2 1 n + 2 n. By the proof of Lemm 3.1, Hn x,y dmits circulr r-colouring c 1 with c 1 (x) = 0 nd c 1 (y) = 1 2. Similrly the grphs Hx,y n,
CIRCULAR COLOURING THE PLANE 5 H y,z n nd H y,z n dmit circulr r-colourings c 2, c 3 nd c 4, respectively, with c 2 (x) = 0, c 2 (y ) = c 4 (y ) = 1 2, c 3(y) = 1 2, nd c 3(z) = c 4 (z) = 0. Now circulr r-colouring c of H n+1 cn e otined y comining the prtil colourings c 1, c 2, c 3, c 4. The construction of this section gives n infinite sugrph of R with circulr chromtic numer t lest 4. It remins open whether or not R hs finite sugrph with the sme property. REFERENCES [1] H. Hdwiger nd H. Derunner, Comintoril geometry in the plne, Holt, Rinehrt nd Winston, New York, 1964. [2] L. Moser nd W. Moser, Solution to prolem 10, Cnd. Mth. Bull. 4 (1961), 187 189. [3] E. R. Scheinermn nd D. H. Ullmn, Frctionl grph theory, John Wiley & Sons Inc., New York, 1997. [4] X. Zhu, Circulr chromtic numer: survey, Discrete Mth. 229 (2001), 371 410.