Multiple Circular Colouring As a Model for Scheduling

Open Journal of Discrete Mathematics, 2013, 3, 162-166 http://dx.doi.org/10.4236/ojdm.2013.33029 Published Online July 2013 (http://www.scirp.org/journal/ojdm)

Multiple Circular Colouring as a Model for Scheduling

Bing Zhou Department of Mathematics, Trent University, Peterborough, Canada Email: [email protected]

Received April 30, 2013; revised May 31, 2013; accepted June 15, 2013

Copyright © 2013 Bing Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT In this article we propose a new model for scheduling periodic tasks. The model is based on a variation of the circular chromatic number, called the multiple circular colouring of the conflict graph. We show that for a large class of graphs, this new model will provide better solutions than the original circular chromatic number. At the same time, it allows us to avoid the difficulty of implementation when the fractional chromatic number is used.

Keywords: Graph Coloring; Circular Chromatic Number; Fractional Chromatic Number; Multi-Circular Coloring; Scheduling Problem

1. Introduction edges between them and therefore can be assigned to one period. For this scheduling f, we have f  1  G . We consider the scheduling problems involving tasks   Thus in general f   1  G . TT12,,, Tn . If two tasks both use a common resource, they cannot be scheduled at the same time. A valid When the tasks are periodic in nature, circular colouring of the conflict graph is a more appropriate method. scheduling is a mapping f from TT12,,, Tn to the subsets of a time period [0,T] such that Circular colouring and the circular chromatic number (also called the star chromatic number) were introduced fTij fT if Ti and Tj use a common resource. Let the value of a scheduling f be by A. Vince in 1988 [5]. A (k,d)-circular-colouring of a graph G is a mapping cVG:0,1,,  k1  such f minfT T :1  i n, which is the minimum i  that for each edge xy in G, dcxcykd . length of time a task has been assigned normalized by the The circular chromatic number of G,  G , is the in- length of the time period. The goal is to find a scheduling c  fimum of the ratio k/d for which G has a (k,d)-circular- f  that maximizes f . One example of this type of colouring. scheduling problem is the heavily loaded resource shar- Equivalently, we can consider a (k,d)-circular-colouring ing system in computer science [1-3]. The tasks are proc- of G as a mapping c from V(G) to the open arcs of length esses and some of them may share a common data file. d in a circle of length k such that if xy is an edge in G, Two processes that do share a common data file cannot operate at the same time. A scheduling of the processes cx   cy    . If we assign the tasks using a (k,d)- that has the maximum value would allow the processes to circular-colouring of the conflict graph G, we would operate most efficiently. have f dk1 c  G . The constraints of the scheduling problem can be rep- A (k,d)-set-colouring of a graph G is a mapping c such resented by a graph, called the conflict graph. The con- that for every vertex v in G, c(v) is a d-subset of 0,1, ,k  1 and if xy is an edge in flict graph G has vertex set vv12,,, vn  representing G,  . The fractional chromatic number of the tasks where two vertices vi and vj are adjacent if and cx  cy    only if they use at least one common resource. Vertex a graph G,  f G , is the infimum of the ratio (k/d) for colouring and chromatic numbers of the conflict graph which G has a (k,d)-set-colouring. If we can consider a have been used as models for scheduling problems (see (k,d)-set-colouring as a mapping c from V(G) to the sets for example [4]). If G can be coloured with kcolours such of arcs in the circle of length k such that the length of the that no two adjacent vertices have the same colour, we union of arcs in f(v) is at least d for every vertex v and if can then divide [0,T] into k equal length periods. All ver- xy is an edge inG, cx   cy  . An assignment using tices that are coloured with the same colour do not have a (k,d)-set-colouring of the conflict graph G would yield

u1 f dk1  f  G . Circular chromatic number and fractional chromatic number and their variations have been extensively stud- v1 ied in the last two decades [6-11]. More results on the u u 5 v v 2 circular chromatic number and fractional chromatic 5 2 number can be found in the book [12] and the survey w v v papers [13,14]. It is easy to see that by their definition, 4 3 we have the relations u4 u3 fcGGG  . Figure 1. Grotzsch graph. Vince proved in [5] that  M t Ct52 2. GG1.c  f   5  Therefore, we have In comparison, we have

GG  c  . t  M Ct5  3. While the difference between  G and c G is always less than 1, it is known that the difference be- For this class of graphs, the difference between  and  f is unbounded. tween  G and  f G can be arbitrarily large for some graphs. An example is the family of graphs called For the circular chromatic number, we have Kneser graphs. A Kneser graph ([15,16]) K has all tt pq, c M CMC55   12t . q-subsets of 0,1,,p  1 (p > 2q) as its vertices and two vertices are adjacent if the two subsets are disjoint. It This shows that the difference between the circular is proved in [16] that chromatic number and the fractional chromatic number is also arbitrarily large for the Mycielskians. To achieve the p  f Kpq,, optimal result for a scheduling problem in general, it q appears that the fractional chromatic number of the con-  Kpq,2  p q 2. flict graph provides the best results. However, there are difficulties if we use (k,d)-set col- Another example of a family of graphs where the dif- ouring and the fractional chromatic number in the sched- ference between their chromatic number and fractional uling problem. To achieve  f G the optimal (k,d)-set chromatic number is large is the graphs obtained by us- colouring may have a very large value of d. As pointed t ing Mycielski’s construction. out in [12], M C5  is an example of a graph G for For a graph G with vertex set VG uu,,, u which  Gkd for no small d. In fact, if we let  12 n  f   and edge set EG, the Mycielskian M G of G is G2 C5 and GMn Gn1  , then the graph with vertex set n1 VG n   32 1 uu12,,, unn vv12 ,,,  v w and edge set

EuvuuEij,:, ij vwii : 1,2,, n. M C5  in and  fnG  is a fraction whose denominator, when 2n2 Figure 1 is also called the Grotzsch graph. written in smallest terms, is greater than 2 . This ex- It is well known that ample shows that there is no bound on the denominator of  f G that is a polynomial function of the number MG G 1. of vertices of G. If this optimal set colouring is to be used For the fractional chromatic number of Mycielskians, for scheduling, each task would be divided into too many the authors of [17] found the recurrence relation fragments making it impossible in practice. To combine optimality and practicality, in the next section we pro- ffM GG 1.fG pose a new colouring of the conflict graph that will provide a better solution than the circular colouring and eas- For the Grotzsch graph, since  C  52, we have f  5  ier to implement than the set colouring.

 f MC5 52 25 2910. 2. Multiple Circular Colouring of a Graph We use the notation M t G such that Definition 1 An m-(k,d)-circular colouring of a graph G t  MG   MM( ( MG ( ))). is a mapping c from V(G) to the sets of open arcs in the t times circle of length k such that c(v) is the union of m arcs We have with a total length at least d and if xy is an edge in G,

Copyright © 2013 SciRes. OJDM 164 B. ZHOU cx cy . The m-circular chromatic number, m c G is the infimum of the ration kd for which Ghas a m-(k,d)-circular-colouring. w u It is easy to see that for every positive integer m, 3 u2 u 4 u GGGm  . 1 fc c w To demonstrate this colouring indeed improves the solution of the scheduling problem in some cases, we 2 show that even for m  2 , c G can be strictly less than c G for large classes of graphs. Recall that for a graph G, M(G) is the Mycielskian of G. There are many graphs G such that

c MG  MG. Some sufficient conditions for this equality to hold are given in, for example, [18] u5 and [19]. However,  2 M G will always be strictly c  Figure 2. A 2-circhular coloring of vertices of C5 and w in M less than .  M G (C5). Theorem 2 For every graph G, v3 2 1 c MG MG . 2 u2

Proof: Let VG uu12,,, un ,  G k and w v1 u3

VMG  uu12,,, unn vv12 ,,,  v w as des- u4 u1 cribed in the previous section. Let 0, k  be a circle of v2 v4 w length k. Since  Gk  , there is a mapping of the form cu i  jj,1 for each i where j0,1, ,k 1 such that cu  cu  when-  ii12  ever u and u are adjacent. v3 i1 i2 v4  1 v  v5 v 2 Let 0, k   be a circle of length k 12. Let 1  2  13 1  u5 cw 0, 1, . Notice that cw cui  22 2 Figure 3. A 2-circular coloring of M (C5). for all i. Figure 2 demonstrates this mapping when

GC 5 . Theorem 4 Let G be a graph of n vertices, and For each i, if cu cw, let cv cu ;  i     ii   m k c G  . Then k and d can be integers such that d < otherwise, let cvii cu  cw  kk,12. We d show that the arcs representing these vertices in the case mn. Proof: Suppose that  m Gr . By scaling if neces- of GC 5 in Figure 3. c   21k  sary, there is a m-circular colouring c that maps the ver- It is easy to check that this is a valid 2- -cir- tices of G to sets of m arcs in a circle of length r such that 2 1 cular colouring of M(G) in general. This proves that each arc has length at least and if xy is an edge then m 2 11 cx cy . We assume that c is a colouring such c MG k  MG .     22 that the set □  1  Corollary 3 If c MG  MG, then ll: is an arc in cv for some vertex v and l  m 2 1 ccMG MG . 2 is the smallest. We fix a direction of the circle, say coun- □ ter clockwise. 1 Next we show that unlike the set-chromatic number, There is at least one arc l such that l  ; otherwise the denominator of the m-circular chromatic number is m bounded by the product of m and the number of vertices r could be made smaller. Let that arc be l1. There must be 1 1 in the graph. an arc l2 such that i) cl1  is adjacent to cl2  in

m G, ii) the left end of l1 is the right end of l2 and iii) dependent set. We have mGc   such independent 1 n l2  , otherwise l1 could be made larger. sets. The average size of one set is m . There- m mGc  Continuing this process, some arc will have to be used fore more than once. Say the first time this happens is at ll . Then the arcs ll,,, l must cover the n ab aa1 b1  G  m . circle an integer number of times. Suppose that they mGc  cover the circle s times. Since there are b a arcs and and 1 n they all have length , we have  m G  . m c mG  ba □ sr  m 3. Conclusion and ba  We proved that for a large class of graphs, this multiple r  sm circular colouring and m-circular chromatic number of the conflict graph will provide better solutions for the Since s  n , the denominator is less than mn. □ scheduling problem than the original circular chromatic Intuitively, when the value of m increases, the m-cir- number. It is also easier to implement than the model cular chromatic number will be closer to the fractional using the fractional chromatic number. We plan to inves- chromatic number and thus the difference between the m tigate more classes of graph G where ccGG    chromatic number and m-circular chromatic number for small values of m. It would also be interesting to find would increase. Nevertheless, our next theorem shows out the probability for random graphs to have m-circular that the m-circular chromatic number cannot be less than chromatic number strictly less than their circular chro- one m-th of the .chromatic number. matic number. 1 Theorem 5 For every graph G, m GG . c m m REFERENCES Proof: Suppose that c Gr  . There is an m-circu- 1 [1] V. C. Barbosa and E. Gafni, “Concurrency in Heavily lar colouringc such that cv for every vertex v. Loaded Neighborhood-Constrained Systems,” ACM Tran- r sactions on Programming Languages and Systems, Vol. Since c(v) is a union of m arcs, at least one of the arcs has 11, No. 4, 1989, pp. 562-584. doi:10.1145/69558.69560 1 [2] H. G. Yeh, “A Connection between Circular Colorin and length at least . We insert mr points PP12,,, Pmr mr Periodic Schedules,” Discrete Applied Mathematics, Vol. spaced at equal distance in the unit length circle. For 157, No. 7, 2009, pp. 1663-1668. every vertex v, c(v) contains at least one of these points. doi:10.1016/j.dam.2008.10.003 [3] H. G. Yeh and X. Zhu, “Resource-Sharing System Sched- Let VvGcvPi:. i Each Vi is an independ- uling and Circular Chromatic Number,” Theoretical Com- ent set and VG    Vi . G can be coloured with mrcol- ours. So we have puter Science, Vol. 332, No. 1-3, 2005, pp. 447-460. doi:10.1016/j.tcs.2004.12.005 m GmrmGc , [4] F. Chung, “Graph Theory in the Information Age,” No- tices of AMS, Vol. 57, No. 6, 2010, pp. 726-732. 1 i.e., m GG . [5] A. Vince, “Star Chromatic Number,” Journal of Graph c m Theory, Vol. 12, No. 4, 1988, pp. 551-559. □ doi:10.1002/jgt.3190120411 The Kneser graphs provide examples showing this [6] H. L. Abbott and B. Zhou, “The Star Chromatic Number lower bound is asymptotically the best possible. of a Graph,” Journal of Graph Theory, Vol. 17, No. 3, Let  G be the independence number of G. 1993, pp. 349-360. doi:10.1002/jgt.3190170309 m [7] H. Hatami and R. Tusserkani, “On the Complexity of the c G is also bounded by a function of  G . The proof above yields a lower bound for  G . Circular Chromatic Number,” Journal of Graph Theory, Vol. 47, No. 3, 2004, pp. 226-230. doi:10.1002/jgt.20022 Theorem 6 For every graph G with n vertices, [8] K. Kilakos and B. Reed, “Fractionally Colouring Total m n c G  . Graphs,” Combinatorica, Vol. 13, No. 4, 1993, pp. 435- mG  440. doi:10.1007/BF01303515

Proof: The set Vi in the proof of Theorem 5 is an in- [9] D. Kral, E. Macajova and J.-S. Sereni, “Circular Edge-

Coloring of Cubic Graphs with Girth Six,” Journal of [15] M. Knser, “Aufgabe 300,” Jahresbericht der Deutsche Combinatorial Theory, Series B, Vol. 100, No. 4, 2010, Mathematiker-Vereinigung, Vol. 58, No. 2, 1955, p. 27. pp. 351-358. doi:10.1016/j.jctb.2009.10.003 [16] L. Lovasz, “Kneser’s Conjecture, Chromatic Number, and [10] I. Leader, “The Fractional Chromatic Number of Infinite Homotopy,” Journal of Combinatorial Theory, Series A, Graphs,” Journal of Graph Theory, Vol. 20, No. 4, 1995, Vol. 25, No. 3, 1978, pp. 319-324. pp. 411-418. doi:10.1002/jgt.3190200404 doi:10.1016/0097-3165(78)90022-5 [11] B. Zhou, “Some Theorems Concerning the Star Chromatic [17] M. Lasen, J. Propp and D. H. Ullman, “The Fractional Number of a Graph,” Journal of Combinatorial Theory, Chromatic Number of Mycielski Graphs,” Journal of Series B, Vol. 70, No. 2, 1997, pp. 245-258. Graph Theory, Vol. 19, No. 3, 1995, pp. 411-416. doi:10.1006/jctb.1996.1738 doi:10.1002/jgt.3190190313 [12] E. R. Scheinerman and D. H. Ullman, “Frational Graph [18] G. Chang, “Circular Chromatic Numbers of Mycielski’s Theory,” John Wiley and Sons, New York, 1997. Graphs,” Discrete Mathematics, Vol. 205, No. 1-3, 205, [13] X. Zhu, “Circular Chromatic Number—A Survey,” Dis- 1999, pp. 23-37. doi:10.1016/S0012-365X(99)00033-3 crete Mathematics, Vol. 229, No. 1-3, 2001, pp. 371-410. [19] H. Jajiaholhassan and X. Zhu, “Circular Chromatic Num- doi:10.1016/S0012-365X(00)00217-X ber and Mycielski Construction,” Journal of Graph The- [14] X. Zhu, “Recent Developments in Circular Colouring of ory, Vol. 44, No. 2, 2003, pp. 106-115. Graphs,” Algorithms and Combinatorics, Vol. 26, Spring- doi:10.1002/jgt.10128 er, Berlin, Heidelberg, 2006, pp. 497-550.