Discrete Mathematics

Transcription

1 Dicrete Mathematic 310 (010) Content lit available at ScienceDirect Dicrete Mathematic journal homepage: Rank number of grid graph Hannah Alpert Department of Mathematic, Univerity of Chicago, 5734 S. Univerity Avenue, Chicago, IL 60637, USA a r t i c l e i n f o a b t r a c t Article hitory: Received 30 Augut 009 Received in revied form 16 July 010 Accepted 1 July 010 Available online 1 Augut 010 Keyword: Vertex ranking Minimal ranking Rank number Grid graph A ranking of a graph i a labeling of the vertice with poitive integer uch that any path between vertice of the ame label contain a vertex of greater label. The rank number of a graph i the mallet poible number of label in a ranking. We find rank number of the Möbiu ladder, K P n, and P 3 P n. We alo find bound for rank number of general grid graph P m P n. 010 Elevier B.V. All right reerved. 1. Introduction Given a graph G, a labeling of the vertice f : V(G) Z + i a ranking if, for every u, v V(G) with f (u) = f (v), every path between u and v contain ome vertex w with f (w) > f (u). We refer to the function value in a ranking a label or color. A ranking i a λ-ranking, for ome λ Z +, if it greatet label i λ. The rank number χ r of G i the leat λ for which G ha a λ-ranking. A χ r -ranking of G i a χ r (G)-ranking. A vertex v 0 of G i a drop vertex of a ranking f if there exit a ranking g with g(v 0 ) < f (v 0 ) and g(v) = f (v) for all vertice v v 0. A ranking i locally minimal if it ha no drop vertex. A ranking f i globally minimal if, for any ranking g, if g(v) f (v) for all v V(G), then g = f. Locally minimal and globally minimal are equivalent [13,11], o both may be called minimal. It i ometime ueful to note that the rank number i the leat number of color in any minimal ranking, a well a in any arbitrary ranking. Throughout thi paper, we ay that a color i in a ranking i high if every color at leat i i aigned to only one vertex, and a vertex i high if it color i high. For any color i in a ranking f of a graph G, let T i denote the et {v V(G) : f (v) > i}. We note that if f i a minimal ranking of G then f retricted to G T i i alo a minimal ranking. One convenient lemma i that if H i a ubgraph of G then χ r (H) χ r (G) [4]. We ue thi lemma repeatedly. The kth power of a path, P k n, i defined to have vertice v 1,..., v n with edge {v i, v j } whenever i j k. Similarly, the kth power of a cycle, C k n, i defined to have vertice v 1,..., v n with edge {v i, v j } whenever i j k or n i j k. The path P n i the ame a P 1 n, and the cycle C n i the ame a C 1 n. The grid graph P m P n i the Carteian product of P m and P n, and the ladder L n i P P n. Rank number were firt defined for tree only [1,], and then were tudied in term of algorithm [1,,4 6,9,10,14, 19]. Ghohal, Lakar, and Pillone defined minimal ranking and began to examine them mathematically [7,8,17,18]. More recent work on ranking ha focued on the arank number, defined to be the larget number of color in any minimal ranking [11,15,16]. addre: hcalpert@uchicago.edu X/$ ee front matter 010 Elevier B.V. All right reerved. doi: /j.dic

2 H. Alpert / Dicrete Mathematic 310 (010) Fig. 1. A 1-bridge and a -bridge, with high vertice hown a open dot. The rank number of P n i χ r (P n ) = log (n) + 1, atifying the recurion χ r (P n ) = 1 + χ r (P n 1 ), for n > 1 []. In [0], Novotny et al. determined the rank number of L n to be n χ r (L n ) = 1 + log n log n 1 + log n atifying the recurion χ r (L n ) = + χ r (L n ), for n >. They alo computed the rank number of the path power P n. Although an early verion of the preent paper included the formula χ r (P k) = n k + χ r(p k n k ) and χ r (C k) = n k + χ r(p k n k ), we omit the proof here becaue they were independently found by Chang et al. and appeared in [3]. In the ame paper, Chang, Kuo, and Lin alo computed the rank number of the prim pri n = P C n, which wa independently found by Ortiz et al. [1]. We ue imilar technique to compute the rank number of the Möbiu ladder möb n obtained by connecting the end of a ladder with a twit, and of K P n. Specifically, χ r (pri n ) = + χ r (L n ) for n > 3, χ r (möb n ) = + χ r (L n 1 ) for n > 3, χ r (K P n ) = χ r(l n ), and χ r (K P n ) = + χ r (K P n ), unle i odd and n atifie a + a 1 n < a + a 1 1 for ome a, in which cae χ r (K P n ) i one more than the value given above. Aymptotically, a n grow large, thi implie that χ r (pri n ) and χ r (möb n ) are log n + O(1), and that χ r (K P n ) = log n + O(1). Novotny, Ortiz, and Narayan alo poed the problem of finding rank number of grid graph P m P n in general. We find the rank number of P 3 P n, howing that it i alway within 3 of 3 log (n). We alo find the following upper bound for rank number of quare grid: for n, χ r (P n P n ) 3n bitcount(n) log n, where bitcount(n) denote the number of 1 in the binary repreentation of n. We find the upper bound χ r (P m P n ) m χ r (P n ), and we find recurive upper and lower bound for χ r (P m P n ) when n i large compared to m.,. The prim and the Möbiu ladder Let pri n denote the prim P C n, and möb n denote the Möbiu ladder, obtained by adding the edge {(u 1, v 1 ), (u, v n )} and {(u, v 1 ), (u 1, v n )} to P P n, where P ha vertice u 1, u and P n ha vertice v 1,..., v n. Recently, Chang et al. [3] and Ortiz et al. [1] independently found the rank number of the prim. The computation of rank number of the prim i included here for comparion with that of the Möbiu ladder. We can eaily ee that möb 3 admit a 4-ranking, but a 4-ranking on pri 3 would require three mutually non-adjacent vertice to be labeled 1, and there are not three uch vertice. Thu χ r (pri 3 ) = 5 and χ r (möb 3 ) = 4. Theorem 1. For n 4, we have χ r (pri n ) = + χ r (L n ) and χ r (möb n ) = + χ r (L n 1 ). Proof. Following [0], we define a 1-bridge to be one rung (coniting of two correponding vertice of the two path) of a ladder, prim, or Möbiu ladder, and a -bridge to be a pair of conecutive rung (coniting of four vertice). We define a 1-bridge to be high if both vertice are high, and a -bridge to be high if two non-adjacent vertice of the bridge are high (Fig. 1). Firt, if G i pri n or möb n, we have χ r (G) + χ r (L n 1 ), becaue we can obtain a ranking of G by adding a high 1-bridge to a χ r -ranking of L n 1. Next, we claim that χ r (G) + χ r (L n ). Let λ equal χ r (L n ) and uppoe that f i a (λ + 1)-ranking of G. Let α be the greatet label that appear more than once. There are two way for T α to eparate the vertice colored α. One way i to put three vertice of T α a the three neighbor of one of the vertice colored α. In that cae, f i a (λ )-ranking on the L ubgraph that doe not contain thoe three neighbor. Thi i impoible becaue rank number of ladder of conecutive length can differ by at mot 1 [0], o χ r (L ) χ r (L n ) 1 = λ 1. The other way for T α to eparate the vertice

3 336 H. Alpert / Dicrete Mathematic 310 (010) colored α i for T α to form two high 1- or -bridge. In that cae, there i an L n 4 ubgraph that exclude thoe bridge, and f retricted to that ubgraph i a (λ 3)-ranking. Thi i impoible becaue (again uing the recurion from [0]) we know that χ r (L n 4 ) = χ r (L n ) = λ. Thu there i no (λ + 1)-ranking of G. We now have the following inequalitie: + χ r (L n ) χ r (pri n ) + χ r (L n 1 ), + χ r (L n ) χ r (möb n ) + χ r (L n 1 ). Thee determine χ r (pri n ) and χ r (möb n ) whenever χ r (L n ) = χ r (L n 1 ). Becaue χ r (L n ) = + χ r (L n ) [0], we ee that the rank number of ladder increae only when the length increae from even to odd length, o, in the remaining cae where χ r (L n ) < χ r (L n 1 ), we mut have n even, for n 4. For thi cae we ue the operation of reduction, defined by Ghohal et al. [7]. If S i a et of mutually non-adjacent vertice of G, then we form the reduction G S of G by S by removing each vertex in S and adding edge o that it neighbor form a clique. If f i a ranking of G S, then g defined by g(v) = 1 for v S and g(v) = f (v) + 1 for v S i a ranking of G, a in [7, Theorem 5]. Now, for n even, C n i the reduction of pri n correponding to a et of n mutually non-adjacent vertice, o, uing the rank number of power of path and cycle given in [3], we have χ r (pri n ) 1 + χ r (C ) = n 3 + χ r(p ) = n + χ r(l n ). Thu χ r (pri n ) = + χ r (L n ). The cae of χ r (möb n ) for χ r (L n ) < χ r (L n 1 ) require a little more work. We will prove ome fact about minimal ranking of ladder, in order to how that no minimal χ r (L n )-ranking of möb n i poible. For any, conider the graph obtained by adding a pendant edge incident to a corner of L to make a new graph H. We note that χ r (H) = χ r (L ) becaue, in any χ r -ranking of L, ome corner vertex w ha label greater than 1, o we can make a χ r -ranking of H by tarting with a χ r -ranking of L, adding a pendant edge incident to w, and coloring the pendant vertex 1. Now we claim that, if χ r (L ) < χ r (L +1 ), then in any minimal χ r -ranking of H, the pendant vertex mut be colored 1. Suppoe not. Then there i a minimal χ r -ranking of H in which the pendant vertex i not colored 1, in which cae it neighbor w i colored 1 [7, Lemma ]. Then we could add one more vertex to make L +1, and color that new vertex 1, becaue the two neighbor of the newet vertex are neighbor of w, o are not colored 1. The reult would be a χ r (L )-ranking of χ r (L +1 ), which i impoible. We ue thi fact to claim by induction that, if χ r (L ) < χ r (L +1 ), then any minimal χ r -ranking of L mut aign 1 to half of the vertice. It i true for = 1 and true for =. For >, any minimal χ r -ranking of L conit of two copie of L (each with a χ r -ranking) and one high -bridge. The deletion of the two high vertice of the bridge reult in two copie of the H of the previou paragraph, each coniting of a pendant edge added to L. Let f be the induced ranking on one of thee graph H. We ee that f i a minimal χ r -ranking. Let u be the pendant vertex in H. Then f (u) = 1, and we claim that f retricted to H u = L i alo minimal. The only way for f not to be minimal on H u i if w, the unique neighbor of u, i a drop vertex and can be recolored 1 in H u. However, in that cae we could obtain a minimal χ r -ranking of H by coloring w a f (u) = 1 and coloring u a f (w). Then the color of u would not be 1, o thi i a contradiction and implie that f retricted to H u i a minimal ranking. Then, by the inductive hypothei, H u = L ha half of the vertice colored 1. Similarly, conider möb n, for χ r (L n ) < χ r (L n 1 ). We will how that any minimal ( + χ r (L n ))-ranking of möb n mut be half 1, and derive a contradiction from the fact that, for n even, möb n doe not contain n mutually non-adjacent vertice. If χ r (L n ) < χ r (L n 1 ), then by the recurion on ladder given in [0] we ee that χ r (L n 4 ) < χ r (L n 4 +1). We ue thi fact to ee that any minimal ( + χ r (L n ))-ranking of möb n mut conit of two copie of L n 4 plu two high -bridge, in order to avoid containing a ranking of L n 4 +1 with only χ r(l n 4 ) color, which would be impoible. The deletion of the two high vertice in one of the -bridge reult in a graph we will call H, with induced minimal χ r (L n )-ranking. Then H conit of L n with two pendant vertice, which we will call u 1 and u and which have neighbor w 1 and w, repectively. Both pendant vertice are colored 1, becaue otherwie w 1 or w i colored 1, in which cae we could add a vertex to make a χ r (L n )-ranking of L n 1, which i impoible. We will how that f retricted to H u 1 i minimal; then the argument of the previou paragraph how that f retricted to H u 1 u i minimal, o H u 1 u i half 1, and thu the whole ranking of möb n i half 1. A before, if f i not minimal on H u 1, then w 1 can be recolored 1 in H u 1. Then we could obtain a minimal χ r -ranking of H by coloring w 1 a f (u 1 ) = 1 and coloring u 1 a f (w 1 ). Then the color of u 1 would not be 1, but we already concluded that the color of u 1 mut be 1 in any minimal χ r -ranking of H. Thu f i minimal on H u 1, o f i half 1 on möb n, giving our contradiction, o χ r (möb n ) = + χ r (L n 1 ). Corollary. A n grow large, we have χ r (pri n ) = log n + O(1), and alo χ r (möb n ) = log n + O(1). 3. The rank number of K P n Now we find the rank number of the graph K P n. We will refer to each copy of K a a layer of K P n, and each copy of P n a a rail.

4 H. Alpert / Dicrete Mathematic 310 (010) Lemma 3. For n 3, we have χ r (K P n ) + χ r (K P n ). Proof. Say that K P n ha vertice (u i, v j ), 1 i, 1 j n. In any λ-ranking of K P n, let α be the greatet label ued more than once. We claim that, in order for T α to eparate the vertice labeled α, there mut be ome j 0 uch that, for all i, either (u i, v j0 ) T α or (u i, v j0 +1) T α. Otherwie, there i a equence {i j } n 1 j=1 uch that neither (u i j, v j ) nor (u ij, v j+1 ) i in T α, and then the equence (u i1, v 1 ), (u i1, v ), (u i, v ), (u i, v 3 ),..., (u in 1, v n 1 ), (u in 1, v n ) i a path through the complement of T α adjacent to every vertex not in T α, o the complement of T α i connected. Thu, there i uch a j 0, and o the λ-ranking of K P n contain a (λ )-ranking of K P n. Lemma 4. χ r (K P ) = +. Proof. Given a ranking of K P, there cannot be two vertice of the ame layer with the ame color, becaue they are adjacent. Let S be the et of color that appear on both layer. Then there can be no rail with both of it vertice colored with color in S. Thi i becaue, if u and v are on uch a rail with f (u) < f (v), then, ince f (v) i in S, there i a vertex w in the layer of u with f (w) = f (v); then w, u, and v together form a bad path, and f would not be a ranking. Thu S contain at mot color, and the remaining vertice are all different color, for a total of + color. Thi can be achieved by chooing rail to have color in S in the firt layer, and chooing other rail to have color in S in the other layer. Theorem 5. Unle i odd and n = 4, for n 3 we have χ r (K P n ) = +χ r (K +P n ). If i odd, then χ r (K P 4 ) = 1+. For the proof we ue two lemma. Firt, we ay that a ranking of K P n ha a connectible end if one of the outer layer ha at mot of the color 1 through. For even, every end i connectible. By the lemma above, for odd we ee that K P ha a χ r -ranking with one connectible end, but none with two connectible end. Lemma 6. If K P n ha a χ r -ranking with a connectible end, then o doe K P n, and χ r (K P n ) = + χ r (K P n ). Proof. The lower bound i in Lemma 3, o we need only exhibit an ( + χ r (K P n ))-ranking. For n odd, take two copie of K P n, find χ r -ranking with connectible end pointing outward, and add a high layer of more color in between. For n even, take two copie of K P n and one copy of K P in the middle. Find χ r -ranking for all of them, with connectible end pointing to the left. Increae the label of the greatet vertice of the middle ection o that they are all high. Then connect the three ection carefully: each two newly conecutive layer mut connect in uch a way that none of the vertice of label 1 through on each layer i adjacent to any of the vertice of label 1 through on the other layer. Thi i poible becaue of the connectible end. It reult in a ranking becaue, if we delete the high vertice in the middle, then each ide conit of a ranking of K P n have arranged them o that the vertice from the middle ection have label at mot with at mot vertice from the middle ection attached. We and their neighbor in the next layer have label greater than. The reulting ranking ue + χr (K P n ) color and ha a connectible end on the left. For even, all end are connectible, o the lemma implie the theorem. Thu, we proceed for odd only. Lemma 7. If i odd, then χ r (K P 4 ) = 1 +. Proof. We want to how that color do not uffice. Suppoe f i a -ranking of K P 4. By the argument in Lemma 3, there are high label retricted to two adjacent layer. They mut be the middle layer, or we would have an impoible -ranking of K P on one ide, and we may aume without lo of generality that thee high label are the greatet label. Then the outer layer mut have exactly the label 1 through. Thu, one of the middle layer hare at leat of the label 1 through with a layer adjacent to it. By the argument in Lemma 4, thi configuration caue a bad path, o f i not a ranking. Thu we need more than color. However, there i a ( + 1)-ranking with connectible end, obtained by uing the color through + 1 on each of the two outer layer, o that they have connectible end, and uing the argument of the previou lemma to connect them to the middle two layer. Proof of Theorem 5. We know for n equal or 4 that K P n ha a χ r -ranking with a connectible end. We how that thi i alo true for n = 3. Lemma 3 implie that χ r (K P 3 ), and it i clear that χ r (K P 3 ), becaue we can chooe the whole middle layer to be high. One poible ranking with connectible end ha 1 through on the left layer, 1 on the middle layer (not adjacent to the 1 on the left, of coure), through on the right layer (none adjacent to the 1), and all the ret high. Thee contitute our bae cae. Applying Lemma 6 complete the proof.

5 338 H. Alpert / Dicrete Mathematic 310 (010) Corollary 8. We have χ r (K P n ) = χ r(l n ), unle i odd and n atifie a + a 1 n < a + a 1 1 for ome a, in which cae χ r (K P n ) i one more than thi value. Corollary 9. A n grow large, we have χ r (K P n ) = log n + O(1). 4. The rank number of P 3 P n In computing χ r (P 3 P n ), we find that it depend on how we can rank P 3 P ; if P 3 P ha a certain property, then it may caue P 3 P n to have a correponding property. Thu, for convenience, we recurively define everal et of natural number; we name them according to their leat element. Cla 7: {7} {b + 3 : b in cla 15}; Cla 15: {a + 1 : a in cla 7}; Cla 6: 1 + cla7; Cla 8: 1 + cla7; Cla 16: 1 + cla15; Cla 17: + cla15; Cla 18: 3 + cla15. Equivalently, number in cla 7 have the form 4 k 7 + 4k 1 5, k 0, and number in cla 15 have the form 3 4 k k 1 7, k 0, with the other clae defined a above. 3 For the ret of thi ection, let r(n) denote χ r (P 3 P n ), and ay that P 3 P n ha three row and n column. It i eay to check that r(1) =, r() = 4, and r(3) = 5. We can how that r(4) = 6 a follow. Suppoe that f i a 5-ranking of P 3 P 4. If there are two vertice labeled 3, then, in order to eparate them, the two vertice labeled 4 and 5 mut be the two neighbor of a corner, inducing a 3-ranking of a P P 3 ubgraph, which i impoible. But if there i only one vertex in each of the color 3, 4, and 5, we find that, after deleting any three vertice from P 3 P 4, the graph till contain a P 4 ubgraph, which doe not admit a -ranking. Theorem 10. For n 5, we have χ r (P 3 P n ) = 3 + χ r (P 3 P ) for n not in cla 16 or 17, and χ r (P 3 P n ) = 4 + χ r (P 3 P ) for n in cla 16 or 17. We will prove the theorem through a equence of lemma. Firt we find a lower bound and then an upper bound that together confine the rank number to a choice of at mot two conecutive number. Then we determine which of thoe two conecutive number i in fact the rank number. We introduce a piece of notation that will be ueful for the coming lemma. In a minimal ranking f of P 3 P n, for n, let α be the greatet label ued more than once. Let C(f ) (C for cut et) denote any mallet ubet of T α with the property that, after the deletion of C(f ), not all vertice labeled α are in the ame connected component. It i clear that C(f ) > 1. If C(f ) =, then C(f ) conit of two neighbor of a corner, and o α = 1 becaue f i minimal. Thu, for α > 1, we have C(f ) 3. Alo note that the vertice in C(f ) are confined to at mot C(f ) conecutive column. Thee obervation imply the following lower bound for r(n). Lemma 11. For n 3, we have r(n) 3 + r( ). Proof. In a minimal λ-ranking f of P 3 P n, for n 3, let α be the greatet label ued more than once. If C(f ) 3, then C(f ) certainly contain three high vertice confined to at mot three conecutive column. If C(f ) =, then α = 1, o it i eay to ee that there are till three high vertice in three conecutive column. Then the complement of thoe three vertice contain P 3 P and i colored with λ 3 color. To retate the lemma, r(n + ) 3 + r(n), for all n 1. Thu, uing the bae cae r() = 4, r(3) = 5, r(4) = 6, we find by induction the lower bound r( k ) 3k, r( k + k ) 3k 1, r( k + k 1 ) 3k, for k. Now we find an upper bound for r(n). Becaue we will need to connect ranking of horter grid together to form ranking of longer grid, we decribe everal type of end that can be attached to P 3 P n to enable the grid to fit together. We ay that a ticky end i three new vertice added to P 3 P n and joined to an outermot column in a tair-tep pattern. So, P 3 P n with two ticky end refer to a graph coniting of P 3 P n plu ix more vertice. For each n there are two uch graph, and they are interchangeable for our purpoe. We exhibit in Fig. a 3-ranking of P 3 P 1 with two ticky end, and a 5-ranking of P 3 P 3 with two ticky end.

6 H. Alpert / Dicrete Mathematic 310 (010) Fig.. A 3-ranking of P 3 P 1 with two ticky end, and a 5-ranking of P 3 P 3 with two ticky end. Fig. 3. How to glue two ticky end, and how to glue a pair end to a pike end, with high vertice hown a open dot. Lemma 1. If there i a λ-ranking of P 3 P n with two ticky end, then there i a (λ + 3)-ranking of P 3 P n+3 with two ticky end. Proof. We glue together two copie of P 3 P n with two ticky end, with a λ-ranking on each copy. The gluing procedure i hown in Fig. 3, which portray the middle column of the new P 3 P n+3. (The figure alo how how to glue a different et of end, ued in a later lemma.) To glue, we add three high vertice between the inward-facing ticky end of the maller ubgraph, o that the three high vertice pan three conecutive column in a diagonal pattern. Thee three conecutive column eparate the copie of P 3 P n, for a total length of n + 3 column, plu two outward-facing ticky end. Uing P 3 P 1 with two ticky end and P 3 P 3 with two ticky end a our bae cae and the lemma above a our inductive tep, we find r( k + k 1 3) 3k 1, r( k+1 3) 3k, for k. Combining thee upper bound with the lower bound given earlier, we find the following bound: 3k r(n) 3k 1, for k n < k + k, r(n) = 3k 1, for k + k n < k + k 1, r(n) = 3k, for k + k 1 n < k+1. We note that the approximation 3 log (n) i alway within 3 of r(n), although thi fact i not relevant to the remainder of the proof. Becaue of the bound above, all that remain i to find out when the rank number increae from 3k to 3k 1. For n > the interval from k to k + k contain one number in cla 7 if k i odd, or one number in cla 15 if k i even. We will how the rank number increae from 3k to 3k 1 at the change from cla 7 to cla 8 and at the change from cla 15 to cla 16. Thi will complete the characterization of r(n), becaue the rank number increae exactly once in each interval from k to k + k, and there i exactly one change from cla 7 to cla 8 or from cla 15 to cla 16 in each uch interval. A pair end i two pendant edge, each incident to a corner of an outermot column of P 3 P n. A pike end i one pendant edge, incident to the middle row of an outermot column. We exhibit in Fig. 4 a 7-ranking of P 3 P 6 with a pike end and a ticky end, and a 7-ranking of P 3 P 7 with two pair end. Lemma 13. For n in cla 6,7, or 15, we have r(n) = 3 + r( ). For n in cla 6, there i an r(n)-ranking of P3 P n with a pike end and a ticky end. For n in cla 7, there a an r(n)-ranking of P 3 P n with two pair end. For n in cla 15, there i an r(n)-ranking of P 3 P n with one pair end and one ticky end, and an r(n)-ranking of P 3 P n with two pike end. Proof. We ue induction, with P 3 P 6 and P 3 P 7 a the bae cae (Fig. 4). The continuing contruction are much like the previou lemma. Here, we may glue two ticky end together with three high vertice in between, o that the ticky end and the high vertice comprie three middle column, or we may glue a pair end to a pike end with three high vertice in between, o that the end and the high vertice comprie two middle column, a hown in Fig. 3. Conider n = a+1 in cla 15, with a in cla 7. Then a 1 i in cla 6, and a 1 =. We claim that r(a) = r(a 1): thi i true for a = 7, and for a > 7 we have a = b + 3 for ome b in cla 15, in which cae r(a) and r(a 1) are both equal to 3 + r(b) by the inductive hypothei. Thu it uffice to make a (3 + r(a))-ranking of P 3 P n with the variou end. To make P 3 P n with a pair end and a ticky end, glue P 3 P a with two pair end to P 3 P a 1 with a pike end and a ticky end. (The pair end and pike end are glued in the middle. In all thee decription, we glue the end lited econd of the firt ubgraph to the end lited firt of the econd ubgraph.) To make P 3 P n with two pike end, glue together P 3 P a 1 with a pike end and a ticky end, to P 3 P a 1 with a ticky end and a pike end.

7 3330 H. Alpert / Dicrete Mathematic 310 (010) Fig. 4. A 7-ranking of P 3 P 6 with a pike end and a ticky end, and a 7-ranking of P 3 P 7 with two pair end. Fig. 5. There i only one 4-ranking of P 3 P with a ticky end. Conider n = b + in cla 6, with b in cla 15. Then = b. To make P3 P n with a pike end and a ticky end, glue together P 3 P b with two pike end, to P 3 P b with a pair end and a ticky end. Conider n = b + 3 in cla 7, with b in cla 15. Again, = b. To make P3 P n with two pair end, glue together P 3 P b with a pair end and a ticky end, to P 3 P b with a ticky end and a pair end. The lemma above implie that, for n in cla 7 or cla 15, and k uch that k n < k + k, then r(m) = 3k for k m n. The following lemma how that r(m) = 3k 1 for n + 1 m < k + k, for n in cla 7 or cla 15, completing the characterization of rank number of P 3 P n for all n. Lemma 14. For n in cla 7 or cla 15, we have r(n) < r(n + 1). Proof. In thi lemma we prove by induction a tronger tatement, a follow: that for n in cla 7 or cla 15 we have r(n) < r(n + 1), that for n in cla 15 there i no r(n)-ranking of P 3 P n with a pike end and a ticky end, that for n in cla 6 there i no r(n)-ranking of P 3 P n with two ticky end, and that for n in cla 7 there i no r(n)-ranking of P 3 P n with a pike end. For the purpoe of thi proof, conider the number to be in cla 15. Note that it atifie 7 = and r() < r(3). For the bae cae, it i eay to check that P 3 P with a ticky end ha only one 4-ranking, up to tranpoing the label 4 and 3. We exhibit thi ranking in Fig. 5, and note that it doe not admit a pike end. We how that, for n in cla 6, there i no r(n)-ranking of P 3 P n with two ticky end. Here, and in the following cae of thi lemma, we have C(f ) 3, a follow. If C(f ) had only two or fewer vertice, for f a minimal χ r -ranking of P 3 P n with variou end for ome n, then we can check that, whatever the two vertice of C(f ) are, their removal reult in two connected component one of which ha rank number at mot. Becaue f i minimal and that mall connected component contain a vertex labeled α (where α a before i the greatet non-high color), we ee that α. However, in the χ r -ranking we are conidering, we have α >, o C(f ) cannot have only two vertice. If n i in cla 6, we can write n a b+, with b in cla 15. Suppoe that there i a minimal r(n)-ranking f of P 3 P n with two ticky end. By the previou lemma, r(n) = 3 + r(b). If C(f ) = 3, we can check each poible configuration of C(f ) to ee that, upon it deletion, one of the reulting connected component contain an r(b)-ranking of P 3 P b with a pike end and a ticky end, which by the inductive hypothei i impoible. If C(f ) > 3, then C(f ) contain four high vertice confined to at mot four conecutive column. Deleting thoe four column leave two connected component, each with an (r(b) 1)-ranking, and one of thee mut contain P 3 P b 1. When n > 6, we know that r(b 1) = r(b), o thi i impoible.

8 H. Alpert / Dicrete Mathematic 310 (010) Fig. 6. Even if no three high vertice eparate P 3 P 6, there i till no 7-ranking with two ticky end. Thu, conider the cae n = 6, and uppoe that there i a minimal 7-ranking f of P 3 P 6 with two ticky end, with C(f ) > 3. If C(f ) contain four high vertice in only three conecutive column, then deleting thee three column leave a 3-ranking of P 3 P, which i impoible. There i a 7-ranking of P 3 P 6 in which C(f ) conit of four high vertice in four column, hown in Fig. 6, but no uch ranking admit even one ticky end. Thi conclude the proof that there i no r(n)-ranking of P 3 P n with two ticky end, for n in cla 6. Now, we how that, if n i in cla 7, there i no r(n)-ranking of P 3 P n with a pike end. Thi will alo how that there i no r(n)-ranking of P 3 P n+1, for n + 1 in cla 8. If n i in cla 7, we can write n a b + 3, with b in cla 15. By the inductive hypothei, P 3 P b+1 require more than r(b) color. In any minimal r(n)-ranking f of P 3 P n, we know that C(f ) contain three high vertice in at mot three conecutive column, and we find that deleting thoe three column leave two copie of P 3 P b, each requiring r(b) color. Then, C(f ) mut be exactly 3, and in order not to contain an r(b)-ranking of P 3 P b+1 (which i impoible), any r(n)-ranking of P 3 P n mut conit of two copie of P 3 P b glued at ticky end to the three vertice of C(f ). Thu, every r(n)-ranking of P 3 P n with a pike end contain an r(b)-ranking of P 3 P b with a pike end and a ticky end, which i impoible by the inductive hypothei. Next, we how that, for n in cla 15, there i no r(n)-ranking of P 3 P n with a pike end and a ticky end. We can write n a a + 1, with a in cla 7. By the inductive hypothei, P 3 P a with a pike end require more than r(a) color, and o doe P 3 P a 1 with two ticky end. Suppoe that there i an r(n)-ranking of P 3 P n with a pike end and a ticky end. We know that r(n) = 3 + r(a). If C(f ) = 3, then we can check the poible configuration of C(f ) to ee that, upon it deletion, the reulting graph contain an r(a)-ranking of either P 3 P a with a pike end, or P 3 P a 1 with two ticky end, both impoible. If C(f ) > 3, then C(f ) contain four high vertice in at mot four conecutive column, and deleting thoe column leave an (r(a) 1)-ranking of a graph containing P 3 P a 1, alo impoible becaue r(a 1) = r(a). Thu there i no r(n)-ranking of P 3 P n with a pike end and a ticky end, for n in cla 15. Finally, we how that there i no r(n)-ranking of P 3 P n+1, for n + 1 in cla 16. We ue the argument in the previou paragraph to ee that, in any r(n)-ranking of P 3 P n+1, we mut have C(f ) = 3. Then we check the poible configuration of C(f ) to ee that, when it i deleted, the reulting graph contain an r(a)-ranking of P 3 P a with a pike end, which i impoible. To conclude the proof of Theorem 10, we need to determine for which n we have r(n) = 4 + r( ). All of thee n have k n < k + k for ome k, and we know that number in thi range greater than cla 7 or cla 15 all have rank number 3k 1, for the appropriate k. Notice that, if n i in cla 18 or greater, then i in cla 8 or greater, o r(n) = 3k 1 = 3+(3(k 1) 1) = 3+r( o r(n) = 3k 1 = 3 + (3(k 1) 1) = 3 + r( Corollary 15. A n grow large, we have χ r (P 3 P n ) = 3 log n + O(1). ). Likewie, if n i in cla 8 or greater, then i in cla 16 or greater, ). Thu, only for n in cla 16 or 17 do we have r(n) = 4 + r( ). 5. Bound for general grid The ubtle characterization of χ r (P 3 P n ) ugget that the rank number of grid are not eaily tated or proven, in general. However, we provide ome bound for the rank number of grid. Theorem 16. For every m, there exit N uch that m + χ r P m P n m χ r (P m P n ) m + χ r for all n > N, and the upper bound hold for all n. P m P n 1 Proof. Say that P m P n ha m row and n column. The upper bound hold for all n, not only large n. To contruct an (m+χ r (P m P n 1 n 1 ))-ranking of P m P n, we color the leftmot column with a χr -ranking of the induced P m P n 1 ubgraph, and color the rightmot n 1, column with a χr -ranking of the induced P m P n 1 ubgraph. Then make all

9 333 H. Alpert / Dicrete Mathematic 310 (010) Fig. 7. How to color tri n, hown for n equal to 6 and 5. The high vertice are hown with big white dot, and the next highet in mall open dot. m vertice in the middle column high. Thu the recurive upper bound χ r (P m P n ) m + χ r (P m P n 1 ) hold, and thi implie the exact upper bound χ r (P m P n ) m ( log (n) + 1) = m χ r (P n ). Now we will how the lower bound. Let f be a minimal λ-ranking of G = P m P n, for n > m. For any color i, let H i be the graph with vertex et T i and edge between vertice that are adjacent or diagonal in the grid. Let α be the greatet label uch that H α contain a connected component of at leat m vertice, and let C be a connected ubgraph of H α containing exactly m vertice. Such an α mn exit becaue H 1 contain at leat vertice if there were mn more than vertice labeled 1, then ome would be adjacent o H1 i connected and contain more than m vertice. Then C pan at mot m column. Let n m A 1 be the induced ubgraph on the leftmot column of Pm P n, and A be the n m induced ubgraph on the rightmot column of Pm P n. Then, becaue there are m 1 or m column in between A 1 and A, we ee that A 1 or A i dijoint from C; call thi graph A. We how that, if n i large, every label greater than α i ued only once. Thu, C ha m different color and f A doe not contain any of thee m color, o f A ue at mot λ m color. In a ranking of any graph G, we ay that T i plit G into iland, where each iland i a connected component of the graph obtained by deleting T i from G. Each iland ha a et of libertie, where a liberty i a vertex of T i adjacent to ome vertex of the iland. Let Q denote the infinite grid graph correponding to one quadrant of the quare lattice in the plane. Let L be the larget poible number of vertice in a finite iland of Q with m libertie. Put N > L+1, o that, if n > N, then χ r (P n ) > L. Conider G = P m P n. We want to how that every label greater than α i ued only once, o uppoe to the contrary that u and v are vertice of T α with f (u) = f (v) = β. Then β > α, o every connected component of H β ha fewer than m vertice. Thi i not enough vertice to reach from the top row of G to the bottom. Thu, when T β plit G into iland, one of the iland contain at leat one vertex from each of the n column, and thu contain P n. Some vertex w on thi path ha L < f (w), becaue of the aumption that χ r (P n ) > L; alo, f (w) β becaue w i not in T β. Thu β > L. Each of the other iland contain at mot L vertice, again becaue H β ha connected component of fewer than m vertice, which do not tretch from top to bottom of G. The hape of thee iland and of their et of libertie could jut a well appear in Q. But, becaue u and v are in different iland, either u or v mut be in one of thee iland of ize at mot L. We aumed that f i a minimal ranking, o f retricted to any iland mut be a minimal ranking, which implie that β L. Thi i a contradiction, which how that no two vertice of label greater than α have the ame label. Becaue C ha m vertice all of which have label greater than α, the fact that A and C are dijoint implie that the ranking on A ue at mot λ m color. Corollary 17. For fixed m a n grow large, we have χ r (P m P n ) = m log n + O(1). Having found bound for long grid, we can alo find an upper bound for quare grid χ r (P n P n ) by offering a trategy for contructing ranking. We tart by making all the vertice on the diagonal high. Deleting thee n vertice leave two triangle, each of which we will denote tri n 1. We will find an upper bound for χ r (tri n 1 ), giving u an upper bound for χ r (P n P n ) becaue χ r (P n P n ) n + χ r (tri n 1 ). Lemma 18. For n 3, we have χ r (tri n ) n 1 + χ r (tri n ). Proof. The coloring trategy i hown in Fig. 7. For n even, we ue the greatet n label to color n vertice acending diagonally from the corner. That leave two pyramid-haped ubgraph that are each n vertice tall. We ue the next n 1

10 H. Alpert / Dicrete Mathematic 310 (010) color to color the center column of each of thoe pyramid, eparating the pyramid into three piece, of which one i a ingle vertex and the other two are tri n. For n odd, we ue the greatet n 1 two pyramid-haped ubgraph, one that i n+1 label to color n 1 vertice acending diagonally from jut above the corner. That leave vertice tall, and one that i n 1 vertice tall. We ue the next n 1 color to or maller. color the center column of each of thoe pyramid, a hown in the figure, leaving piece that are tri n 1 Corollary 19. For n, we have χ r (P n P n ) 3n bitcount(n) log n. 6. Concluion To complete the determination of rank number of grid graph would require finding a good lower bound for rank number of quare grid. There are alo other natural extenion of the theorem in thi paper. Jut a we can connect the end of K P n to make pri n and möb n, we can connect the end of K P n. There are many way to do o, one for each conjugacy cla of S. Similarly, we can connect the end of P 3 P n to make either P 3 C n or another Möbiu-type graph. Another intereting extenion i to conider parallelogram ubet of the hexagonal lattice, a an analogue of grid graph, which are rectangular ubet of the quare lattice. Thee parallelogram are generalization of P n, much a grid graph are a generalization of ladder. Acknowledgement Thi reearch wa upervied by Joe Gallian at the Univerity of Minneota Duluth, upported by the National Science Foundation and the Department of Defene (grant number DMS ) and the National Security Agency (grant number H ). I alo want to thank Nathan Kaplan, Melanie Matchett Wood, Nathan Pflueger, and the referee for editing my draft. Reference [1] H.L. Bodlaender, J.S. Deogun, K. Janen, T. Klok, D. Kratch, H. Müller, Z. Tuza, Ranking of graph, in: Graph-Theoretic Concept in Computer Science (Herrching, 1994), in: Lecture Note in Comput. Sci., vol. 903, Springer, Berlin, 1995, pp [] H.L. Bodlaender, J.S. Deogun, K. Janen, T. Klok, D. Kratch, H. Müller, Z. Tuza, Ranking of graph, SIAM J. Dicrete Math. 11 (1) (1998) (electronic). [3] C.-W. Chang, D. Kuo, H.-C. Lin, Ranking number of graph, Inform. Proce. Lett. 110 (16) (010) [4] J.S. Deogun, T. Klok, D. Kratch, H. Müller, On vertex ranking for permutation and other graph, in: STACS 94, Caen, 1994, in: Lecture Note in Comput. Sci., vol. 775, Springer, Berlin, 1994, pp [5] J.S. Deogun, T. Klok, D. Kratch, H. Müller, On the vertex ranking problem for trapezoid, circular-arc and other graph, Dicrete Appl. Math. 98 (1 ) (1999) [6] D. Dereniowki, A. Nadolki, Vertex ranking of chordal graph and weighted tree, Inform. Proce. Lett. 98 (3) (006) [7] J. Ghohal, R. Lakar, D. Pillone, Minimal ranking, Network 8 (1) (1996) [8] J. Ghohal, R. Lakar, D. Pillone, Further reult on minimal ranking, Ar Combin. 5 (1999) [9] S. Hieh, On vertex ranking of a tarlike graph, Inform. Proce. Lett. 8 (3) (00) [10] R. Hung, Optimal vertex ranking of block graph, Inform. and Comput. 06 (11) (008) [11] G. Iaak, R. Jamion, D. Narayan, Greedy ranking and arank number, Inform. Proce. Lett. 109 (15) (009) [1] A.V. Iyer, H.D. Ratliff, G. Vijayan, Optimal node ranking of tree, Inform. Proce. Lett. 8 (5) (1988) 5 9. [13] R. Jamion, Coloring parameter aociated with ranking of graph, in: Proceeding of the Thirty-Fourth Southeatern International Conference on Combinatoric, Graph Theory and Computing, vol. 164, 003, pp [14] T. Klok, H. Müller, C.K. Wong, Vertex ranking of ateroidal triple-free graph, Inform. Proce. Lett. 68 (4) (1998) [15] V. Kotyuk, D. Narayan, Maximum minimal k-ranking of cycle, Ar Combin. (009) (in pre). [16] V. Kotyuk, D. Narayan, V. William, Minimal ranking and the arank number of a path, Dicrete Math. 306 (16) (006) [17] R. Lakar, D. Pillone, Theoretical and complexity reult for minimal ranking, J. Combin. Inform. Sytem Sci. 5 (1 4) (000) Recent advance in interdiciplinary mathematic (Portland, ME, 1997). [18] R. Lakar, D. Pillone, Extremal reult in ranking, in: Proceeding of the Thirty-Second Southeatern International Conference on Combinatoric, Graph Theory and Computing, Baton Rouge, LA, 001, vol. 149, 001, pp [19] C. Liu, M. Yu, An optimal parallel algorithm for node ranking of cograph, Dicrete Appl. Math. 87 (1 3) (1998) [0] S. Novotny, J. Ortiz, D. Narayan, Minimal k-ranking and the rank number of P n, Inform. Proce. Lett. 109 (3) (009) [1] J. Ortiz, H. King, A. Zemke, D. Narayan, Minimal k-ranking of prim graph, Involve (009) (in pre). [] A. Schäffer, Optimal node ranking of tree in linear time, Inform. Proce. Lett. 33 () (1989)