Fibonacci Identities and Graph Colorings

FIBONACCI IDENTITI ES AND GRAP H COLO RINGS CHRISTOPHER J. HILLAR AND TROELS WINDFELDT Abstra ct. W e generalize b oth the Fib onacci and Lucas num b ers to the co ntext of graph colorings, and prov e some iden tities in vol ving these num bers. As a corollary we obtain new pro ofs of some known identities inv olving Fib onacci num b ers such as F r + s + t = F r +1 F s +1 F t +1 + F r F s F t − F r − 1 F s − 1 F t − 1 . 1. Introduction In graph theory , it is natural to stud y v ertex co lorings, and more sp ecifi- cally , those colorings in wh ic h adj acen t v er tices ha ve different colors. In this case, the num b er of such colorings of a grap h G is enco ded by th e c hromatic p olynomial of G . Th is ob j ect ca n b e compu ted using the metho d of “dele- tion and contract ion”, w hic h in v olv es the recursive com bination of c hromatic p olynomials for sm aller graphs. The p urp ose of this note is to show ho w the Fib onacci and L u cas num b ers (and other intege r recurrences) arise naturally in this con text, and in particular, ho w iden tities amo ng these num b ers c an b e generated from the different c hoices for decomp osing a graph into smaller pieces. W e first in tro duce some notation. Let G b e a u ndirected graph (p ossibly con taining lo ops and multiple edges) w ith vertice s V = { 1 , . . . , n } and edges E . Giv en non n egativ e in tegers k and ℓ , a ( k, ℓ ) -c oloring of G is a map ϕ : V − → { c 1 , . . . , c k + ℓ } , in which { c 1 , . . . , c k + ℓ } is a fixed set of k + ℓ “colors”. The map ϕ is called pr op er if whenev er i is adjacen t to j and ϕ ( i ) , ϕ ( j ) ∈ { c 1 , . . . , c k } , w e ha v e ϕ ( i ) 6 = ϕ ( j ). Otherwise, we sa y that the map ϕ is impr op er . In somewhat lo oser terminology , one can think of { c k +1 , . . . , c k + ℓ } as coloring “wildcards”. Let χ G ( x, y ) b e a f unction such that χ G ( k , ℓ ) is the num b er of prop er ( k , ℓ )-colorings of G . This ob ject w as in tro duced by the authors of [DPT03] and can b e giv en as a p olynomial in x an d y (see Lemma 1.1). It sim ultane- ously generalizes th e c hromatic, ind ep enden ce, an d matc hing p olynomials of 1991 Mathematics Subje ct Cl assific ation. 05C15, 05A19, 05C38. Key wor ds and phr ases. Graph colorings, Fib onacci identities, chromatic polyn omial, linear recurrences. The first author is supp orted u nder a N ational Science F oun d ation Postdoctoral F ellowship. 1 2 CHRISTOPHER J. HILLAR AND TROELS WINDFELDT G . F or instance, χ G ( x, 0) is the usu al c hromatic p olynomial wh ile χ G ( x, 1) is the ind ep end en ce p olynomial for G (see [D PT03] for more detai ls). W e next s tate a simp le rule that enables one to calculate the p olynomial χ G ( x, y ) recursiv ely . In what follo ws, G \ e d enotes the graph obtained by remo ving the edge e from G , and for a su b graph H of G , the graph G \ H is gotten from G by remo ving H and all the edges of G that are adjacent to vertic es of H . Additionally , the c ontr action of an edge e in G is the graph G/e obtained b y remo ving e and iden tifying as equal the tw o v ertices sharing this edge. Lemma 1.1. L e t e b e an e dge in G , and let v b e the vertex to which e c ontr acts in G/ e . Then, χ G ( x, y ) = χ G \ e ( x, y ) − χ G/e ( x, y ) + y · χ ( G/e ) \ v ( x, y ) . (1.1) Pr o of. The num b er of prop er ( k , ℓ )-colorings of G \ e which ha v e distinct colors f or the vertic es sh arin g edge e is giv en by χ G \ e ( k , ℓ ) − χ G/e ( k , ℓ ); these colorings a re al so prop er for G . The remaining pr op er ( k , ℓ )-colorings of G are precisely those for wh ic h th e vertic es sharing edge e ha v e the same color. This color must b e one of the w ildcards { c k +1 , . . . , c k + ℓ } , and so the n umber of remaining prop er ( k , ℓ )-colorings of G is c oun ted by ℓ · χ ( G/e ) \ v ( k , ℓ ).  With such a recur r ence, we n eed to sp ecify initial conditions. When G simply consists of one v ertex and has no edges, we ha v e χ G ( x, y ) = x + y , and when G is the empty graph, we set χ G ( x, y ) = 1 (consider G with one edge joining tw o vertice s in (1.1)). Moreo ver, χ is multiplica tiv e on disconnected comp onent s. This allo ws us to compute χ G for an y graph recur s iv ely . In the sp ecial case when k = 1, there is also a wa y to calculate χ G (1 , y ) by remo ving v ertices fr om G . Defin e the link of a v ertex v to b e the su bgraph link( v ) of G co nsisting of v , the edges touc hing v , and the vertic es sharin g one of these edges with v . Also if u and v are joined by an edge e , we define lin k( e ) to b e link( u ) ∪ link( v ) in G , and also w e set deg( e ) to b e deg( u ) + deg( v ) − 2. W e then ha v e the f ollo wing rules. Lemma 1.2. L et v b e any vertex o f G , and let e b e any e dge. Then, χ G (1 , y ) = y · χ G \ v (1 , y ) + y deg( v ) · χ G \ link( v ) (1 , y ) , (1.2) χ G (1 , y ) = χ G \ e (1 , y ) − y deg( e ) · χ G \ link( e ) (1 , y ) . (1.3) Pr o of. The num b er of prop er (1 , ℓ )-co lorings of G with vertex v colored with a wildcard is ℓ · χ G \ v (1 , ℓ ). Moreo ve r, in an y prop er coloring of G with v colored c 1 , eac h v ertex among the deg ( v ) ones adjacent to v can only b e one of the ℓ w ild cards. This explains th e fir s t equalit y in the lemma. Let v b e the v ertex to whic h e contrac ts in G/e . F rom equation (1.2), we ha v e χ G/e (1 , y ) = y · χ ( G/e ) \ v (1 , y ) + y deg( v ) · χ ( G/e ) \ link( v ) (1 , y ) . FIBONACCI IDENTITIES AND G RAPH COLORINGS 3 Subtracting this equation from (1 .1) with x = 1, and n oting that deg ( e ) = deg( v ) and G \ link ( e ) = ( G/e ) \ link( v ), w e arr iv e at th e second equalit y in the lemma.  Let P n b e the path grap h on n vertice s and let C n b e the cycle graph , also on n v ertices ( C 1 is a v ertex w ith a loop attac h ed while C 2 is t w o v ertices joined by t wo edges). Fixing nonn egativ e intege rs k and ℓ not b oth zero, w e define the follo wing sequences of n umb ers ( n ≥ 1): a n = χ P n ( k , ℓ ) , b n = χ C n ( k , ℓ ) . (1.4) As we shall see, these num b ers are n atural generalizations of b oth the Fi- b onacci and Lucas n umb ers to the co n text of graph colo rings. The follo w- ing lemma uses graph decomp osition to giv e sim p le recurrences for these sequences. Lemma 1.3. The se q uenc es a n and b n satisfy the fol lowing line ar r e cur- r enc es with initia l c onditions: a 1 = k + ℓ, a 2 = ( k + ℓ ) 2 − k , a n = ( k + ℓ − 1) a n − 1 + ℓa n − 2 ; (1.5) b 1 = ℓ, b 2 = ( k + ℓ ) 2 − k , b 3 = a 3 − b 2 + ℓa 1 , (1.6) b n = ( k + ℓ − 2) b n − 1 + ( k + 2 ℓ − 1) b n − 2 + ℓb n − 3 . (1.7) Mor e over, the se quenc e b n satisfies a shorter r e curr enc e if and only if k = 0 , k = 1 , or ℓ = 0 . When k = 0 , this r e curr enc e is give n by b n = ℓb n − 1 , and when k = 1 , it is b n = ℓb n − 1 + ℓb n − 2 . (1.8) Pr o of. The first recur r ence follo ws from deleti ng an outer ed ge of the path graph P n and usin g Lemma 1.1. T o v erify th e second one, we first use Lemma 1.1 (picking an y edge in C n ) to give b n = a n − b n − 1 + ℓa n − 2 . (1.9) Let c n = b n + b n − 1 = a n + ℓa n − 2 and notice that c n satisfies the same recurrence as a n ; namely , c n = a n + ℓa n − 2 = ( k + ℓ − 1) a n − 1 + ℓa n − 2 + ℓ (( k + ℓ − 1) a n − 3 + ℓa n − 4 ) = ( k + ℓ − 1)( a n − 1 + ℓa n − 3 ) + ℓ ( a n − 2 + ℓa n − 4 ) = ( k + ℓ − 1) c n − 1 + ℓc n − 2 . (1.10) It follo ws t hat b n satisfies the third order recurrence giv en in the statemen t of the lemma. Additionally , the in itial conditions for b oth s equ ences a n and b n are easily work ed o ut to be the ones sho wn . Finally , su pp ose that th e sequence b n satisfies a shorter r ecurrence, b n + r b n − 1 + sb n − 2 = 0 , 4 CHRISTOPHER J. HILLAR AND TROELS WINDFELDT and let B =   b 3 b 2 b 1 b 4 b 3 b 2 b 5 b 4 b 3   . Since the n onzero v ector [1 , r , s ] T is in th e k ernel of B , w e m ust hav e t hat 0 = d et( B ) = − k 2 ( k − 1) ℓ (( k + ℓ − 1) 2 + 4 ℓ ) . It follo ws that for b n to satisfy a s m aller recurrence, w e must ha v e k = 0, k = 1, or ℓ = 0. It is clear that wh en k = 0, we ha v e b n = ℓ n = ℓb n − 1 . When k = 1, w e can use Lemma 1.2 to see t hat b n +1 = ℓ ( a n + ℓa n − 2 ) , and com bining th is with (1.9) giv es the recurr en ce stated in the lemma.  When k = 1 and ℓ = 1, the recurrences giv en by Lemm a 1.3 when ap- plied to the families of path graphs and cycle graph s are the Fib onacci and Lucas n umb ers, resp ectiv ely . This observ ation is well -kno wn (see [Kos01, Examples 4.1 and 5.3]) a nd was brought to our at ten tion by Co x [Co x 07 ]: χ P n (1 , 1) = F n +2 and χ C n (1 , 1) = L n . (1.11) Moreo ve r, when k = 2 and ℓ = 1, the recurrence give n by Lemma 1.3 when applied to the family of p ath graphs is the one associated to the P ell n um b ers: χ P n (2 , 1) = Q n +1 , where Q 0 = 1, Q 1 = 1, and Q n = 2 Q n − 1 + Q n − 2 . 2. Iden tities In th is section, we derive some iden tities in v olving the generalized Fi- b onacci a nd Lucas num b ers a n and b n using th e grap h coloring in terpr eta- tion found her e. In what follo ws, we fix k = 1. In this ca se, the a n and b n satisfy the follo wing rec urrences: a n = ℓa n − 1 + ℓa n − 2 and b n = ℓb n − 1 + ℓb n − 2 . Theorem 2.1. The fol lowing identities hold : b n = ℓa n − 1 + ℓ 2 a n − 3 , (2.1) b n = a n − ℓ 2 a n − 4 , (2.2) a r + s = ℓa r a s − 1 + ℓ 2 a r − 1 a s − 2 , (2.3) a r + s = a r a s − ℓ 2 a r − 2 a s − 2 , (2.4) a r + s + t +1 = ℓa r a s a t + ℓ 3 a r − 1 a s − 1 a t − 1 − ℓ 4 a r − 2 a s − 2 a t − 2 . (2.5) Pr o of. All the iden tities in the statemen t of the theorem follo w fr om Lemma 1.2 when applied to differen t graphs (with certa in choice s of ver- tices and edges). T o see the first t w o equations, consider the cycle grap h C n and pic k an y v ertex and an y ed ge. T o see the next tw o equations, consider the path graph P r + s with v = r + 1 and e = { r , r + 1 } . FIBONACCI IDENTITIES AND G RAPH COLORINGS 5 r vertic es t v ertice s s v ertices e v In order to pro v e the final equation in the statmen t of th e theorem, consider the graph G in the ab o ve figure. It foll o w s fr om Lemma 1.2 th at ℓa r + s a t + ℓ 3 a r − 1 a s − 1 a t − 1 = a r + s + t +1 − ℓ 4 a r − 2 a s − 2 a t − 1 . Rearranging the terms and applying (2.4), we see that a r + s + t +1 = ℓa r + s a t + ℓ 3 a r − 1 a s − 1 a t − 1 + ℓ 4 a r − 2 a s − 2 a t − 1 = ℓ ( a r a s − ℓ 2 a r − 2 a s − 2 ) a t + ℓ 3 a r − 1 a s − 1 a t − 1 + ℓ 4 a r − 2 a s − 2 a t − 1 = ℓa r a s a t − ℓ 3 a r − 2 a s − 2 ( ℓa t − 1 + ℓa t − 2 ) + ℓ 3 a r − 1 a s − 1 a t − 1 + ℓ 4 a r − 2 a s − 2 a t − 1 = ℓa r a s a t + ℓ 3 a r − 1 a s − 1 a t − 1 − ℓ 4 a r − 2 a s − 2 a t − 2 . This completes the p ro of of t he theorem.  Corollary 2.2. The fol lowing identities ho ld: L n = F n +1 + F n − 1 , L n = F n +2 − F n − 2 , F r + s = F r +1 F s + F r F s − 1 , F r + s = F r +1 F s +1 − F r − 1 F s − 1 , F r + s + t = F r +1 F s +1 F t +1 + F r F s F t − F r − 1 F s − 1 F t − 1 . Pr o of. The indenti ties follo w from th e corr esp onding ones in Theorem 2.1 with ℓ = 1 by m aking suitable shifts of the indices and us ing (1.11).  3. Fur ther Explora tion In this note, w e hav e pro du ced r ecurrences and iden tities by decomp osing differen t c lasses of graphs in different wa ys. O ur treatment is b y no means exhaustiv e, and there sh ould b e many w a ys to expand on what w e hav e d one here. F or instance, is there a graph coloring proof of C assini’s ident it y? Referen ces [Co x07] Da vid Cox, Pri vate c ommunic ation, 2007. [DPT03] Klaus Dohmen, Andr´ e P¨ onitz, and Pe ter Tittmann, A new two-variable gen- er alization o f the chr omatic p olynomial , Discrete Math. Theor. Comput. Sci. 6 (2003), no. 1, 69–89 (electronic). MR 1996108 (2004j:0 5053) [Kos01] Thomas Koshy , Fib onac ci and Luc as numb ers with applic ations , Pure and Applied Mathematics (New Y ork), Wiley-I ntersci ence, N ew Y ork, 2001. MR 1855020 (2002f:110 15) 6 CHRISTOPHER J. HILLAR AND TROELS WINDFELDT Dep ar tmen t of Ma thema ti cs, Texas A&M University, College St a tion, TX 77843, USA . E-mail addr ess : chi llar@math.t amu.edu Dep ar tmen t of Ma the ma tical Sciences, Universi ty of Copen hagen, De n- mark. E-mail addr ess : win dfeldt@math .ku.dk