The Local Harmonious Chromatic Problem

Transcription

1 The 7th Workshop o Combiatorial Mathematics ad Computatio Theory The Local Harmoious Chromatic Problem Yue Li Wag 1,, Tsog Wuu Li ad Li Yua Wag 1 Departmet of Iformatio Maagemet, Natioal Taiwa Uiversity of Sciece ad Techology, Taipei, Taiwa Departmet of Computer Sciece ad Iformatio Maagemet, Soochow Uiversity, Taipei, Taiwa Abstract The harmoious chromatic umber of graph G, deoted h(g), is the least umber of colors which ca be used to color V (G) such that adjacet vertices are colored differetly ad each color-pair occurs o the vertices of a edge at most oce. I this paper, we geeralize the above problem to be the local harmoious chromatic problem. The local harmoious chromatic problem restricts that the differet color-pair requiremet is oly asked to be satisfied for every edge withi distace d for ay vertex. 1 Itroductio I [7], Frak, Harary, ad Platholt defied that a k-colorig of a graph G is a map f : V (G) C such that C = k. If f(v) = c 1 ad f(w) = c for adjacet v ad w, we say that the color-pair {c 1, c } is preset at the edge vw. A k-colorig is lie-distiguishig if o color-pair is preset at more tha oe edge, ad the liedistiguishig chromatic umber of G, deoted λ(g), is the miimum k such that G has a liedistiguishig k-colorig. Hopcroft ad Krishamoorthy [10] idepedetly studied the above problem ad called λ(g) the harmoious colorig umber. They called the problem to determie whether λ(g) k the harmoious colorig problem ad proved that the harmoious colorig problem is NP-complete. This work was supported i part by the Natioal Sciece Coucil of the Republic of Chia uder the cotract NSC 97-1-E MY3. Correspodig author. I [1], Miller ad Pritiki added oe more coditio to these colorigs, amely that adjacet vertices receive distict colors ad called the problem the harmoious chromatic problem. I the appedix of [10], David S. Johso already gave a very simple proof of the NP-completeess of the harmoious chromatic problem. I this paper, we follow Miller ad Pritiki s defiitio. Specifically, a harmoious k-colorig of graph G is a k- colorig with adjacet vertices receivig differet colors ad all edges receivig differet color-pairs. The harmoious chromatic umber of G, deoted h(g), is the miimum k for which G has a harmoious k-colorig. Determiig whether G ca be colored harmoiously by k colors is also NPcomplete o trees [5] ad iterval graphs []. I this paper, we geeralize the harmoious colorig problem to be the local harmoious chromatic problem. The local harmoious chromatic problem restricts the differet color-pair requiremet oly eeded to be satisfied for every edge withi distace d for ay vertex. Formally, a k- colorig is a d-harmoious colorig if every colorpair is preset at most oce for every edge withi distace d of ay vertex v V (G), ad the d-harmoious chromatic umber of G, deoted h d (G), is the miimum k such that G has a d- harmoious colorig. Clearly, the origial harmoious chromatic problem is the -harmoious chromatic problem if V (G) = ; or, more precisely, the x/ -harmoious chromatic problem if x is the diameter of G. The rest of this paper is orgaized as follows. I Sectio, we prove that the 1-harmoious chromatic problem for geeral graphs is NP-complete. I Sectio 3, we itroduce how to fid the d- harmoious chromatic umber of paths. I Sec- 37

2 The 7th Workshop o Combiatorial Mathematics ad Computatio Theory tios ad 5, we are cocered with the d- harmoious chromatic umber of cycles for ay positive iteger d. Fially, our cocludig remarks are give i Sectio 6. Local harmoious chromatic umbers of paths I this paper, we cosider oly simple graphs G with vertex set V (G) ad edge set E(G). For defiitios of graph theoretic terms see Harary [9]. Sice the harmoious colorig problem is NPcomplete, determiig whether h (G) k is also NP-complete. I the followig, we prove that the 1-harmoious chromatic problem, i.e., determiig whether h 1 (G) k, is also NP-complete. Theorem 1. The 1-harmoious chromatic problem for geeral graphs is NP-complete. I the rest of this sectio, we shall derive the d- harmoious chromatic umber for a path P, i.e., h d (P ). The followig two propositios will be used to derive h d (P ). Propositio. A graph G is d-harmoious colorig if ad oly if o color-pair appears more tha oce for every subpath P d+1 i G. Propositio 3. Assume that x is the diameter of G. The h d (G) = h(g) for every d x/. As metioed i [7], Frak et al. used a trail o a complete graph to determie the harmoious colors of P ad C. For example, the Euleria circuit 1,,3,,5,1,3,5,,,1 of K 5 as show i Figure 1(a) ca be used to color the edges of C 10 (see Figure 1(b)) ad P 11 (see Figure 1(c)). { Accordigly, they derived } λ(p ) = mi, { ad λ(c ) = mi }, I the followig, we use lec(g) to stad for the legth of a Euleria subgraph which has the maximum possible umber of edges i E(G). It is obvious that lec(k ) = ( ( ) if is odd; ad ) otherwise. By usig a similar techique, Miller ad Pritiki [1] proposed the followig theorem for calculatig h(p ). Theorem (Miller ad Pritiki [1]). h(p ) = { k for all lec(kk 1 ) < 1 lec(k k ) + 1 k + 1 for all lec(k k ) + 1 < 1 lec(k k+1 ). 3 1 (a) K (c) P (b) C 10 Figure 1: Colorig paths or cycles with a aided Euler circuit. We ca also use a similar techique to fid h d (P ) as the followig theorem. Theorem 5. If d + 1, the h d (P ) = h(p ); otherwise, h d (P ) = { k for all lec(kk 1 ) < d lec(k k ) k + 1 for all lec(k k ) < d lec(k k+1 ). 3 Periodical tables I this sectio, we are cocered with the d- harmoious chromatic umber of C. First we defie some terms which will be used later. A graph is pacyclic if it has cycles of every legth ragig from 3 to [1, 3, 8, 13]. Clearly, K is pacyclic. We exted the defiitio of pacyclic to x-pacircuitous. A graph is a x-pacircuitous graph if it has circuits of every legth ragig from 3 to x. The reaso why addig a x before pacircuitous is that, eve for K, there are o T lec(k) 1 ad T lec(k) i K if is a odd umber. If is eve, the there do ot exist circuits of every legth ragig from ( ) ( + 1 to ). To emphasize that Clec(K) 1 ad C lec(k) caot be d-harmoiously colored by colors, these two umbers lec(k ) 1 ad lec(k ) are called the first -pump umber ad the secod -pump umber, respectively, for odd iteger 5 (or -pump umbers for short). For example, umbers 8 ad 9 are the secod ad the first, respectively, 5-pump umbers while 19 ad 0 are the secod ad the first, respectively, 7- pump umbers. We use the followig propositio to describe the abset circuits i K whe is odd. Propositio 6. If ( 5) is a odd iteger, the K has o T x where x is the first or the secod -pump umber

3 The 7th Workshop o Combiatorial Mathematics ad Computatio Theory Note that if a graph is x-pacircuitous, x > 3, the it must be (x 1)-pacircuitous. However, by Propositio 6, it might ot be (x+1)-pacircuitous eve if x + 1 < ( ). The pacircuit umber of G, deoted p(g), is the maximum k such that G is k-pacircuitous. For iteger d 1, a clique K k is said to be the critical clique with respect to d, abbreviated as d-critical clique, if it is the clique with the smallest k such that K k has T d+1. I the rest of this paper, we use K α to deote the d-critical clique. Lemma 7 describes the value for p(k ) while Lemma 8 gives a ecessary ad sufficiet coditio for a d-critical clique which is ot (d + 1)-pacircuitous. Lemma 7. For a complete graph K,, { lec(k ) 3 if is odd p(k ) = lec(k ) if is eve. Lemma 8. For a d-critical clique K α with d, K α is ot (d+1)-pacircuitous if ad oly if α = (1 + 16d + 9)/ is a odd umber. Sice there exist pump umbers, h(c ) caot be computed exactly the same as the computatio of h(p ). After takig ito cosideratio of pump umbers, we ca obtai the followig theorem. Theorem 9. If lec(k k 1 ) < lec(k k ) + 1, the h(c ) = k; otherwise, h(c ) is equal to { k + 1 if is ot a (k + 1)-pump umber, k + if is a (k + 1)-pump umber. Lemma 10. For 3 d 1, h d (C ) = h(c ). Corollary 11. For d ad d < d 1, h d (C +1 ) h d (C ) 1. Lemma 1. If K α is the d-critical clique, the lec(k α ) d 1. By Lemma 10, fidig h d (C ), for d, ca be trasformed to the problem of determiig whether ca be composed of umbers x with d x d 1 such that the maximum h d (C x ) amog all compositio umbers is miimum. For brevity, whe sayig the legth of a cycle is composed of the legths of aother two cycles, we simply say that a cycle ca be composed of aother two cycles. For ease of descriptio, we defie a periodical table as follows. Give a iteger d, a periodical table with respect to d, deoted PT d, is a table of d rows ad a ifiite umber of colums. Etry (i, j) of PT d, deoted PT d (i, j), cotais cycle C with = jd + i 1. Sice every = jd + i 1 with itegers j > 1 ad 1 i d ca be expressed as = (j 1)d+(d+i 1), ca be composed of d ad d + i 1. This gives a trivial upper boud of h d (C ) max{h d (C d ), h d (C d+i 1 )} for d. To fid the exact value of h d (C ) for d, we eed to fid a compositio of which results i the exact value of h d (C ). For example, see Table 1 for illustratig a periodical table with d = 3. The etries i the first colum of Table 1 are PT 3 (1, 1), PT 3 (, 1),..., PT 3 (6, 1) which cotai C 6, C 7,..., C 11, respectively. By Lemma 10, all h 3 (C ) = h(c ) for 6 11 which are show i the first colum of Table 1. Sice all of umbers = 1, 18,..., are multiples of d(= 6), h 3 (C ) = 5 for = 1, 18,.... Number 1 ca be composed of by usig oly 7, i.e., 1 = The, by repeatig the colors for C 7 twice o C 1, we ca obtai h 3 (C 1 ) = h 3 (C 7 ) = 5. Note that 1 is also equal to 8 + 6; however, i this compositio, max{h 3 (C 8 ), h 3 (C 6 )} = max{6, } = 6 which is ot the smallest value for h 3 (C 1 ). The other compositios for cycles i colum of Table 1 are 13 = 7+6, 15 = 9+6, 16 = 10+6, ad 17 = 10+7 whose correspodig h 3 (C ) are show i the secod colum of Table 1. Note that 15 ca be composed either of or 8 + 7; however, both of these two compositios result i h 3 (C 15 ) = 6. I the third colum of Table 1, 1 = ad h 3 (C 1 ) = 5. The other values of i colum 3, i.e., PT 3 (i, 3) with = 3 d + i 1, ca be expressed as = d+( d+i 1), which is a compositio of d(= 6) ad ( d + i 1). Note that d + i 1 represets the etry PT 3 (i, ) which is the previous etry i the same row of PT 3 (i, 3). Similarly, we ca obtai h 3 (C ) for cycles i colum of Table 1. Note that, by addig d to the idex of each cycle i the fourth colum, we ca obtai the cycles i the fifth colum of Table 1. Thus, their h d (C ) ca also be foud. Similarly, we ca fid h d (C ) for the remaiig cycles. From the above aalysis, we ca fid h 3 (C ) for 3 as the followig theorem. Theorem 13. if = 3, For 3, h 3 (C ) = 6 if = 8, 9, 11, 15 5 otherwise. Note that PT 1 (1, 1) results i = ad o cycle of legth. This reveals that the case where d = 1 is a special case i fidig h d (C ) by usig the periodical table. Therefore, we cosider this case separately i the followig theorem ad assume that d i the rest of this paper. 39

4 The 7th Workshop o Combiatorial Mathematics ad Computatio Theory Table 1: Periodical table h 3 (C ) for 6 9 h 3 (C 6 ) = 5 h 3 (C 1 ) = 5 h 3 (C 18 ) = 5 h 3 (C ) = 5 h 3 (C 7 ) = 5 h 3 (C 13 ) = 5 h 3 (C 19 ) = 5 h 3 (C 5 ) = 5 h 3 (C 8 ) = 6 h 3 (C 1 ) = 5 h 3 (C 0 ) = 5 h 3 (C 6 ) = 5 h 3 (C 9 ) = 6 h 3 (C 15 ) = 6 h 3 (C 1 ) = 5 h 3 (C 7 ) = 5 h 3 (C 10 ) = 5 h 3 (C 16 ) = 5 h 3 (C ) = 5 h 3 (C 8 ) = 5 h 3 (C 11 ) = 6 h 3 (C 17 ) = 5 h 3 (C 3 ) = 5 h 3 (C 9 ) = 5 Theorem 1. 3 if = 3i, for ay iteger i 1 h 1 (C ) = 5 if = 5 otherwise. Lemma 15 describes a basic property o the compositio of cycle legths i periodical tables. Lemma 15. Let C x be the cycle i PT d (i, j) for itegers i, j 1. The, there exist a pair of cycles C y ad C z which are i PT d (m, 1) ad PT d (i m+1, j 1), respectively, for ay 1 m i, such that x = y + z. Local harmoious chromatic umbers of cycles Based o the periodical table, we shall discuss the compositio of umber for d uder the d-harmoious costrait i order to fid h d (C ). Recall that K α is the d-critical clique. By Corollary 11, the possible values of h d (C d ) is α 1, α, ad α + 1. Accordig to the possible values of h d (C d ) ad the parity of α, we classify periodical tables with respect to d ito the followig eight classes: (1) h d (C d ) = α 1 ad d + 1 is the secod (α 1)-pump umber, () h d (C d ) = α 1, α is eve, ad lec(k α 1 ) = d, (3) h d (C d ) = α 1 ad α is odd, () h d (C d ) = α ad d + 1 is the first (α 1)- pump umber, (5) h d (C d ) = α, α is eve, ad d + 1 is ot the first (α 1)-pump umber, (6) h d (C d ) = α, α is odd, ad d + is ot the secod α-pump umber, (7) h d (C d ) = α ad d + is the secod α-pump umber, (8) h d (C d ) = α + 1 ad α is odd, which are deoted by PT 1 d, PT d,..., PT 8 d, respectively. Note that the case where h d (C d ) = α + 1 with eve α is impossible sice h d (C d ) = α + 1 must be a pump umber which exists oly for odd α. We use the followig propositios to describe some properties of these eight classes of periodical tables i which we assume that K α is the d-critical clique. Propositio 16. For the cycles i the first colum of a PT d PT 1 d, oly h d (C d ) = h d (C d+3 ) = α 1. All of the other cycles, say C, i the first colum of PT d have h d (C ) α. Propositio 17. For the cycles i the first colum of a PT d PT d PT 3 d, oly h d (C d ) = α 1. All of the other cycles, say C, i the first colum of PT d have h d (C ) α. Propositio 18. For the cycles i the first colum of a PT d PT d, oly h d (C d+ ) = α 1. All of the other cycles, say C, i the first colum of PT d have h d (C ) α. Propositio 19. For the cycles i the first colum of a PT d PT 5 d PT 6 d PT 7 d, all cycles, say C, i the first colum of PT d have h d (C ) α. Furthermore, there are cycles whose legths are α-pump umbers i the first colum of a PT d PT 6 d PT 7 d. Propositio 0. For the cycles i the first colum of a PT d PT 8 d, oly h d (C d+1 ) = α. All of the other cycles, say C, i the first colum of PT d have h d (C ) α + 1. I a periodical table, we defie that the ith-level leap path for i 1, is a Mahatta path which starts from the edge separatig C lec(kα+i 1) ad C lec(kα+i 1)+1, the passes through the edges separatig C j lec(kα+i 1) ad C j lec(kα+i 1)+1, for j >, oe by oe, util it arrives the bottom edge of the table. Note that, oce o C j lec(kα+i 1) exists i the secod colum of PT d, the ith-level leap path oly cotais the bottom edge of the table which is the last leap path. For coveiece, we say that the top edge of a periodical table is the 0th-level leap path. Therefore, there are at least two leap paths, i.e., the top ad bottom edges, i a periodical table. Based o leap paths, we defie leap cycles as follows. A cycle C with d is called a 0

5 The 7th Workshop o Combiatorial Mathematics ad Computatio Theory (d, i)-leap cycle, for i 0, if C is located betwee the ith-level ad the (i + 1)th-level leap paths i PT d. Formally, a cycle C i PT d (i, j) is a (d, 0)-leap cycle if ad oly if j ad 1 i mi{d, j (lec(k α ) d) + 1}, ad is a (d, x)-leap cycle, for 1 x k, if ad oly if j ad j (lec(k α+x 1 ) d) + 1 < i mi{d, j (lec(k α+x ) d) + 1} where k is the umber of leap paths. A (d, x)-leap cycle C, for x 0, i PT d (i, j) is called a (d, x)-pump cycle if = j lec(k α+x ) y, for y = 1 or, ad lec(k α+x ) y is a (α+x)-pump umber. A (d, 0)- leap cycle C is called a type I ripple cycle if it is i a PT d PT 7 d ad d is the first α-pump umber. A (d, 0)-leap cycle C is called a type II ripple cycle if it is i a PT d PT 8 d ad C is ot a (d, 0)-pump cycle. Sice all cycles have to be composed of the cycles i the first colum of a PT d, the cycles C i the first colum of PT d with h d (C ) = α + x, for x 0, are called (d, x)-fudametal cycles. For simplicity, we also classify cycles C i the first colum of a PT d with h d (C ) = α 1 ito (d, 0)-fudametal cycles. Lemma 1 ca be used to determie whether there exist (d, x)-leap cycles or ot for a give d ad x 1. Lemma 1. Assume that K α is the d-critical clique ad x( 1) is a positive iteger. If lec(k α+x 1 ) < (6d 1)/, the there exist (d, x)- leap cycles. I the followig, Lemmas 9 cocer with the d-harmoious chromatic umber of (d, 0)-leap cycles ad Lemmas 30 ad 31 cocer with the d- harmoious chromatic umber of (d, x)-leap cycles for PT x d, x = 1,,..., 8. Lemma. For a PT d PT 1 d, a (d, 0)-leap cycle C, which is i PT d (i, j), has h d (C ) = α 1 if ad oly if it has i = 3x+1 for x = 0, 1,..., d 1/3. Lemma 3. For a PT d PT d PT 3 d, a (d, 0)- leap cycle C i PT d has h d (C ) = α 1 if ad oly if = dj for j. Lemma. For a PT d PT d, a (d, 0)-leap cycle C has h d (C ) = α 1 if ad oly if = (d+)j for j. Lemma 5. Assume that K α is the d-critical clique ad C is a (d, 0)-leap cycle of PT d 1 x 5 PTx d. If h d (C ) α 1, the h d (C ) = α. Lemma 6. For a (d, 0)-leap cycle C of PT d PT 6 d, if C is a (d, 0)-pump cycle, the h d (C ) = α + 1; otherwise, h d (C ) = α. Propositio 7. For a (d, 0)-leap cycle C of PT d PT 7 d, C is a type I ripple cycle if ad oly if C is i PT d (j, j) for j (d+5)/. Lemma 8. For a (d, 0)-leap cycle C of PT d i PT 7 d, if C is a (d, 0)-pump cycle or a type I ripple cycle for j, the h d (C ) = α + 1; otherwise, h d (C ) = α. Lemma 9. For a (d, 0)-leap cycle C of PT d i PT 8 d, if = (d+1)j, for j, the h d (C ) = α; otherwise, h d (C ) = α + 1. By usig a similar argumet as Lemma 6 for provig the d-harmoious chromatic umber of leap cycles, we ca obtai Lemmas 30 ad 31. Lemma 30. Assume that K α is the d-critical clique. If C is a (d, x)-leap cycle, for x 1, which is ot a (d, x)-pump cycle, the h d (C ) = α + x. Lemma 31. Assume that K α is the d-critical clique. If C is a (d, x)-pump cycle for x 1, the h d (C ) α + x + 1. For coveiece to summarize our results, a (d, x)-leap cycle for x 0, is called a (d, x)-perfect cycle if h d (C ) = α+x 1, is called a (d, x)-critical cycle if h d (C ) = α+x, ad is called a (d, x)-pump cycle if h d (C ) = α + x + 1. Note that perfect cycles ca oly exist i (d, 0)-leap cycles which are described i Lemmas, critical cycles are described i Lemmas 5 30, ad the existece of (d, x)-pump cycles is described i Lemma 1. We summarize our results as the followig theorem. Theorem 3. Assume that K α is a d-critical clique with d. For ay C, if C is a (d, x)- pump cycle or types I ad II ripple cycles, the h d (C ) = α + x + 1; otherwise, h d (C ) is equal to h(c ) if 3 d 1 α 1 if C is a (d, 0)-perfect cycle α + x if C is a (d, x)-critical cycle. 5 Coclusio Here we wat to poit out that h d (C ) might be much larger tha α + where K α is a d-critical clique. For example, K 65 is a 1000-critical clique, amely d = 1000 ad α = 65. Sice d 1 = 3999, h 1000 (C 3999 ) = h(c 3999 ) = 91 α +. Eve though fidig h 1 (G) for geeral graphs is NPcomplete, however, for a lot of classes of graphs, there exist polyomial time algorithms for fidig their h 1 (G). For example, h 1 (T ) = + 1 where T is a tree. Accordigly, it is iterestig to fid the value of h (G) for trees ad other classes of graphs. 1

6 The 7th Workshop o Combiatorial Mathematics ad Computatio Theory Refereces [1] B. Alspach, Cycles of each legth i regular touramets, Caadia Mathematical Bulleti, 10 (1967) [] K. Asdre, K. Ioaidou, ad S. D. Nikolopoulos, The harmoious colorig problem is NP-complete for iterval ad permutatio graphs, Discrete Applied Mathematics 155 (007) [3] J. A. Body, Cycles of each legth i regular touramets, Joural of Combiatorial Theory, Series B, 11 (1971) [] K. Edwards ad C. McDiarmid, New upper bouds o harmoious colorigs, Joural of Graph Theory, 18 (199) [5] K. Edwards ad C. McDiarmid, The complexity of harmoious colourig for trees, Discrete Applied Mathematics, 57 (1995) [6] K. Edwards, The harmoious chromatic umber ad the achromatic umber, I: R.A. Bailey, ed., Surveys i Combiatorics 1997 (Ivited papers for 16th British Combiatorial Coferece) (Cambridge Uiversity Press, Cambridge, 1997) [7] O. Frak, F. Harary, ad M. Platholt, The lie-distiguishig chromatic umber of a graph, Ars Combiatoria, 1 (198) 1 5. [8] F. Harary ad L. Moser, The theory of roud robi touramets, The America Mathematical Mothly, 73 (1966) [9] F. Harary, Graph Theory (Addiso-Wesley, Readig, MA, 1969). [10] J. E. Hopcroft ad M. S. Krishamoorthy, O the harmoious colorig of graphs, SIAM Joural o Algebraic Discrete Methods, (1983) [11] Zhikag Lu, The harmoious chromatic umber of a complete biary ad triary tree, Discrete Mathematics, 118 (1993) [1] Z. Miller ad D. Pritiki, The harmoious colorig umber of a graph, Discrete Mathematics, 93 (1991) [13] J. W. Moo, O subtouramets of a touramet, Caadia Mathematical Bulleti, 9 (1966)