Partial Chromatic Polynomials and Diagonally Distinct Sudoku Squares

P artial Chromatic P olynomials and Diagonally Distinct Sudoku Squares F ¨ usun Akman Il li nois State University, Normal, IL akmanf@ilstu.edu August 21, 2018 1 In tro duction This paper is based on a talk I ga ve a t Illinois State University on April 10, 2008, and contains t wo pro ofs. The fir st one is of a statemen t a bout comple- tions o f partial λ -colorings of a gra ph in a v ery in teresting article by Herzb erg and Mur t y [2], namely , the fact that the num b er of p ossible completions is a po lynomial in λ (which we will call the p artial chr omatic p olynomial ). Two elegant pro ofs of this statemen t, o ne with M¨ obius in version and the o ther by induction, a r e already giv en in [2]: b oth pr oofs use the co ncept of con trac tion. Our alternative pr oof mimics the constructio n of the classic a l chromatic po ly- nomial instead. The second proof in this paper shows that there exist n 2 × n 2 Sudoku squares with distinct entries in bo th diagonals (in addition to the r o ws , columns, and n × n sub-gr ids) for all n . I w ould lik e to thank W alter “W al” W allis for pointing out (da ys after I p osted the proo f and gav e the talk ) that there exists a n earlier and very similar pro of o f the existence of s uch squar es, due to A.D. Ke e dwell [3]: I was unaw are of [3] at the time. I would also like to thank Papa Sissokho for correcting my terminolo gy and making the firs t pro of more palatable. 2 P artial c hr omati c p olynomial Sudoku puzzles are, in a discrete mathematician’s world, partially colored g r aphs. Questions a bout the minimal num b er of clues fo r unique solutions etc. b oil down to questions ab out par tial coloring s of the “ Sudoku graph” . This graph c onsists of n 4 vertices, corresp onding to the sq uares o f a n n 2 × n 2 Sudoku grid, such that any tw o distinct v e r tices in the sa me row, column, or sub-grid are joined by a n edge. A completed Sudoku puzzle is then a prop er colo ring of the Sudoku graph with n 2 colors. Theorem 1. [2] L et G b e a gr aph with n vertic es, and C b e a p artial pr op er c oloring of t vertic es of G using exactly λ 0 c olors. Define p G,C ( λ ) to b e t he 1 numb er of ways C c an b e c omplete d to a pr op er λ -c oloring of G . Then for λ ≥ λ 0 , the expr ession p G,C ( λ ) is a monic p olynomial in λ of de gr e e n − t . Pr o of. Let C be a partial pr o per color ing of t vertices of G with exactly λ 0 colors. Call a prop er coloring C ′ of G “ consisten t with C ” if the vertices colo red under C keep their colors under C ′ . A lso call a pro per coloring C ′ “generic” if it is simply a partitioning of the v ertices of G into independent sets (more precisely , a gener ic coloring is an equiv a lence class of colorings with the same independent sets). Now let C ′ be any generic prop e r coloring of G with exactly λ 0 independent sets. If C ′ is c onsisten t with C , then there is o nly 1 w ay the colors of C ′ can b e sp ecified; the large r indep endent sets in C ′ hav e to r etain the co lors of the smaller ones in C . Next, if a generic C ′ is to b e consistent with C and have λ 0 + 1 independent sets, then ther e are ( λ − λ 0 ) ways o f sp ecifying the colo rs of C ′ : for the λ 0 sets tha t extend those in C , w e ha ve no choice but to resp ect the colo rs dictated b y C . O n the other hand, the ex tr a indep endent set do es not intersect C , so w e can use any of the remaining ( λ − λ 0 ) colors. W e contin ue the argument for all generic prop er colorings with ex a ctly λ 0 + r independent sets, where 0 ≤ r ≤ n − t . In shor t, we have p G,C ( λ ) = n − t X r =0 m r ( G, C )( λ − λ 0 ) · · · ( λ − λ 0 − r + 1) . Here m r ( G, C ) is the n umber of g e ne r ic pro p er color ings C ′ of G that are co n- sistent w ith C and have ex a ctly λ 0 + r indep enden t sets, and ( λ − λ 0 ) · · · ( λ − λ 0 − r + 1) is the n umber of wa ys the colors of suc h C ′ can b e specified. The r -th ter m o f the sum is a p olynomial of degree r , and the ( n − t )-th term is monic, b ecause there is only one generic C ′ that a dds n − t indep enden t sets to C . Namely , each v er tex outside C is a set by itself. 3 Diagonally distinct Sudoku squares The existence o f n 2 × n 2 Sudoku s q uares for any p ositive integer n is a well-known fact (see [2] for a pro of ). W e will show that it is mor eo ver p ossible to construct n 2 × n 2 Sudoku s quares with distinct entries on each of the tw o diago nals for any n . A similar pro of was given e arlier, and unknown to the a uthor at the time of e-publication of the first version of this paper, by Keedwell [3]. Michalowski et al. [4] a nd Bailey et al. [1 ] give so me motiv ating real-life examples for v ariations of Sudo ku puzzle s and o ther ger ec hte designs. Theorem 2. Ther e exist n 2 × n 2 Sudoku squar es with distinct entries in the two diagonal s, in addition to distinct entries in e ach r ow, c olumn , and n × n sub-grid. Pr o of. Notation: the ( i, j )- blo ck will be the n × n sub-grid placed ac c o rding to matrix-entry enumeration co n ven tion inside the full grid ( i th fr o m the top a nd j th fro m the left). W e will also e numerate entries in any blo c k b y the r ow and 2 column num b ers in the blo c k; thus, the ( r , c )-e ntry of the ( i, j )-blo c k will b e the ( r + ( i − 1) n, c + ( j − 1) n )-entry of the complete gr id. When writing indices, we will a lw ays ch o ose the lea st p ositive residue mo dulo n (denoted by [ x ] for any int e g er x ). As a result, all v ar iables i, j, r , c, [ x ] will have v alues in the set { 1 , . . . , n } . Let the symbols a ( r, c ), with 1 ≤ r, c ≤ n , denote the n 2 int eg ers from 1 to n 2 in so me order . W e place these distinct in teg ers in the upp er le ft n × n blo c k of the grid, no w called the (1 , 1 )-block, such that a ( r, c ) is in r o w r and column c : a (1 , 1) a (1 , 2) a (1 , 3) a (2 , 1) a (2 , 2) a (2 , 3) a (3 , 1) a (3 , 2) a (3 , 3) = 1 2 3 4 5 6 7 8 9 In or der to c r eate the (1 , 2 )-block, we simply move the rows of the (1 , 1)-blo c k up in a cyclic fashion: a (2 , 1) a (2 , 2) a (2 , 3) a (3 , 1) a (3 , 2) a (3 , 3) a (1 , 1) a (1 , 2) a (1 , 3) = 4 5 6 7 8 9 1 2 3 W e contin ue this permutation of r o ws inside each new blo c k until we finish the first row of blo cks. As for the (2 , 1)-blo ck, we a dv ance the rows inside the (1 , 1)- blo c k one step down cy c lic ally , and also mov e the entries in each row (inside the blo c k) one step backward: a (3 , 2) a (3 , 3) a (3 , 1) a (1 , 2) a (1 , 3) a (1 , 1) a (2 , 2) a (2 , 3) a (2 , 1) = 8 9 7 2 3 1 5 6 4 W e co mplete the second row of blo cks similar to the first, only by p erm uting whole rows in the (2 , 1)-blo ck upw ard, without making any c ha ng es to the rows int er nally , and rep eat the pro cedure un til all rows of blo c k s ar e exhausted. The 4 × 4, 9 × 9, a nd 16 × 16 Sudoku sq uares with distinct dia g onal entries constructed by this metho d are given be low: 1 2 3 4 3 4 1 2 4 3 2 1 2 1 4 3 3 1 2 3 4 5 6 7 8 9 4 5 6 7 8 9 1 2 3 7 8 9 1 2 3 4 5 6 8 9 7 2 3 1 5 6 4 2 3 1 5 6 4 8 9 7 5 6 4 8 9 7 2 3 1 6 4 5 9 7 8 3 1 2 9 7 8 3 1 2 6 4 5 3 1 2 6 4 5 9 7 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 9 1 0 11 12 13 14 15 16 1 2 3 4 5 6 7 8 13 14 15 16 1 2 3 4 5 6 7 8 9 1 0 11 12 14 15 16 13 2 3 4 1 6 7 8 5 10 11 12 9 2 3 4 1 6 7 8 5 10 1 1 12 9 14 15 16 13 6 7 8 5 10 11 1 2 9 14 15 16 13 2 3 4 1 10 11 12 9 14 15 16 13 2 3 4 1 6 7 8 5 11 12 9 10 15 16 13 14 3 4 1 2 7 8 5 6 15 16 13 14 3 4 1 2 7 8 5 6 11 12 9 10 3 4 1 2 7 8 5 6 11 1 2 9 1 0 1 5 16 13 14 7 8 5 6 11 12 9 10 15 16 13 14 3 4 1 2 8 5 6 7 12 9 10 11 16 13 14 15 4 1 2 3 12 9 10 11 16 13 14 15 4 1 2 3 8 5 6 7 16 13 14 15 4 1 2 3 8 5 6 7 12 9 10 11 4 1 2 3 8 5 6 7 12 9 10 11 16 13 14 15 W e now present the full pr oof of ex istence. Let us plac e the integer a ( [ r − ( i − 1) + ( j − 1) ] , [ c + ( i − 1) ] ) = a ( [ r − i + j ] , [ c + i − 1] ) in the ( r , c )- en try of the ( i, j )-blo ck, and use the prime nota tio n to distinguish another entry . Distinct en trie s in the same row o f the full grid (where i = i ′ and r = r ′ , but j 6 = j ′ or c 6 = c ′ ) are not equal: if they were, then w e would hav e [ r − i + j ] = [ r − i + j ′ ] and [ c + i − 1] = [ c ′ + i − 1] ⇒ j = j ′ and c = c ′ . Similarly , distinct entries in the same column of the full matrix (where j = j ′ and c = c ′ , but i 6 = i ′ or r 6 = r ′ ) cannot b e equal: [ r − i + j ] = [ r ′ − i ′ + j ] and [ c + i − 1] = [ c + i ′ − 1 ] ⇒ i = i ′ and r = r ′ . 4 Two distinct en tr ies in the same block (wher e i = i ′ and j = j ′ , but r 6 = r ′ or c 6 = c ′ ) are not equal: [ r − i + j ] = [ r ′ − i + j ] and [ c + i − 1] = [ c ′ + i − 1] ⇒ r = r ′ and c = c ′ . Two distinct ent r ies on the main diagonal (where i = j , i ′ = j ′ , r = c , and r ′ = c ′ , but i 6 = i ′ or r 6 = r ′ ) are not equal: [ r − i + i ] = [ r ′ − i ′ + i ′ ] and [ r + i − 1] = [ r ′ + i ′ − 1 ] ⇒ r = r ′ and i = i ′ . Finally , tw o distinct en tr ies on the s econdary dia g onal (wher e i + j = i ′ + j ′ = n + 1, r + c = r ′ + c ′ = n + 1 , but i 6 = i ′ or r 6 = r ′ ) are not equa l: [ r − i + ( n + 1) − i ] = [ r ′ − i ′ + ( n + 1) − i ′ ] and [( n + 1) − r + i − 1 ] = [( n + 1 ) − r ′ + i ′ − 1 ] ⇒ [ r − 2 i ] = [ r ′ − 2 i ′ ] and [ − r + i ] = [ − r ′ + i ′ ] ⇒ r = r ′ and i = i ′ . Calculations o f the sy mmetries, the n umber of essentially different squares, the minimum n umber of entries in a puzzle to assure a unique solution, the asymptotic v alues of related express ions, and the par tial or full chromatic p oly- nomials for Sudoku gr a phs o f rank n , a re mentioned in [2] in relation to standard Sudoku s quares. Similar calcula tions would certainly be interesting for the di- agonally distinct n 2 × n 2 Sudoku squar es. References [1] R.A. Ba iley , P .J . Camer on, and R. Connelly , Sudoku, gerechte de- signs, resolutio ns , affine space , spreads, reguli, and Hamming co des, Amer. Math. Monthly (May 2008). [2] A.M. Her z b erg and M.R. Murty , Sudoku squares and chromatic poly no mi- als, Notic es of the AMS 54(6) (200 7 ), 708-7 1 7. [3] A.D. K eedw ell, On Sudoku s quares, Bul letin of the ICA 50 (20 0 7), 52 -60. [4] M. Michalo ws ki, C.A. Knoblo c k, and B.Y. Choueiry , h ttp://cons ystlab . unl.edu/our ∗ work/Pap ers/Michalo wskiCPWS0 7.p df (insert the unders c ore character for ∗ ) 5