What the theorem says, roughly speaking, is that if you have a zero of a polynomial, then you have a factor 6 factor theorem
In this case, The Remainder Theorem tells us the remainder when p(x) is divided by (x - c), namely p(c), is 0, which means (x - c) is a factor of p What we
Corollary (The Factor Theorem) A polynomial f(x) has (x b a) as a factor if and only if f(a) = 0 The Remainder Theorem follows immediately from the definition
We define a graph as a set V of objects called vertices together with a set E of objects called edges, the two sets having no common element With each edge
In this section, we will learn to use the remainder and factor theorems to factorise and to solve polynomials that are of degree higher than 2 Before doing so,
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THEORE M C. G is without an f-factor if and only if there is a subset S of V and a subset T of V - S such that (6 ) T,f(a)<Theore m C ; detail s ar e give n i n (5) . However , proof s o f Theore m C , eve n i n th e specia l cas e dealin g wit h ^-factors , hav e hithert o bee n lon g an d complicate d (1 ; 5). In this paper we present a comparatively short argument whereby Theore
m C i s deduce d a s a consequenc e o f Theore m A .Deductio
n o f Theore m C fro m Theore m A . Suppos e firs t tha t G has a verte x a suc h tha t d{a) < f{a). The n G can have no/-factor. Moreover (6) is satisfie d wit h 5 = 0 an d T = {a}. Thu s Theore m C i s triviall y tru e i n thi s case . I n th e remainin g cas e w e hav e d(a) > f(a) fo r eac h aÇ F. We write s (a) = d(a) - f{a). Give n an y sufficientl y larg e se t Q w e defin e a grap h G' whos e vertice s ar e element s o f Q i n th e followin g way . Wit h eac h c (z V we associate d(c) distinct element s cA of Q, one for each edge A of G such that e(A, c) = 1, and s(c) other distinc t element s c(l) , c(2) , . . . , c(s(c)) o f Q. W e denot e th e set s o f th e d(c) element s cA and the s{c) elements c(i) by X(c) and Y(c) respectively. WeTHE FACTOR THEOREM FOR FINITE GRAPHS 349
postulat e tha t th e tw o set s X(c) U Y(c) define d fo r tw o distinc t element s c o f V shal l hav e n o commo n element . Th e se t V o f vertice s o f G' i s give n b y (7 ) V = U (X(c){JY(c)). ceV Fo r an y edg e A o f G , wit h end s x an d y say, we postulate that G' has just on e edg e joinin g xA and yA. We denote this also by the symbol A. We further postulat e tha t fo r eac h c Ç V each element of X(c) is joined to each member of Y(c) b y jus t on e edg e o f G' , an d tha t G' ha s n o edge s othe r tha n thos e require d b y thes e tw o rules . Fo r eac h c G V, the elements of X(c) U Y(c) and the edges of G' joining the m constitut e a subgraph , St(c) , o f G' , whic h w e cal l th e star-graph of c in G'. St(c ) i s connecte d i f s(c) > 0 , an d i n th e cas e s(c) = 0 onl y i f d(c) = f(c) = 1 . Th e diagra m show s a star-grap h St(c ) fo r th e cas e d(c) = 4 an d f(c) = 2 . (Th e edge s ABC an d D i n thi s diagra m d o no t belon g t o St(c). ) A A D 6 CG C C|
LEMMA . G has anf-factor if and only if G' has a \-factor. Proof. If G has an /-factor let F be its set of edges and let Ff be the set of edge s o f Gf denoted by the same letters. For each c G V we adjoin to Ff exactly s(c) edge s joinin g th e s(c) vertice s o f Y(c) t o th e s(c) vertice s o f X(c) whic h ar e no t end s o f edge s o f Ff. By the definition of Gf we can do this without intro ducin g int o F' tw o edge s wit h a commo n end . W e thu s construc t a 1-factor of G' .Conversel
y suppos e G' ha s a 1-factor whose set of edges is H. Let Ho be the se t o f edge s o f H whose two ends are vertices of distinct star-graphs St(c). Fo r eac h c G V just s(c) elements of H have an end in Y(c) and therefore just d(c) - s(c) = f(c) element s o f H0 have an end in X(c). It follows that the edge s o f G corresponding to the members of Ho define an /-factor of G.350 W. T. TUTTE
A subse t W o f V wil l b e calle d simple i f i t satisfie s th e followin g condition s fo r eac h a G V: (i ) I f X(a) nW^O the n X(a) Ç W, (ii ) I f Y (a) Pi W 9* 0 then 7(a) C IF, (iii ) A t mos t on e o f X(a) an d Y (a) is a subset of W.Conditio
n (iii ) implie s tha t X(a) canno t b e a subse t o f W whe n Y (a) is the nul l set , i.e. , whe n d(a) = f(a).Conside
r an y simpl e subse t W o f V. W e writ e 5 an d T fo r th e set s o f vertice s c o f G such that X(c) QW and Y(c) C H^ respectively. The sets 5 and T are disjoint . W e hav e (8 ) a(W) = Z (d(c) -f(c)) + £ d(a). ceT aeS Le t H be any component of G'w. I t ma y happe n tha t H has just one vertex, which is of the form cA. Then c G T and the end of A in G other than c belongs to S. The number of such component s H is the number of edges A of G having one end in S and the other i n r , tha t i s ^ 2 ^ (d(c) - ds(c)).Anothe
r possibilit y i s tha t H has just one vertex, which is of the form c(i). Th e numbe r o f suc h component s i sZ(d(a)-f(a)). aeS
I n th e remainin g case , H has at least one edge. If H has no edge in common wit h on e o f th e star-graph s St (a) it must consist of a single edge with its two ends . Then the number of vertices of H is even. If H has an edge in common wit h S t (a) then Y (a) ^ 0 and so St (a) is connected. Moreover St (a) is then a subgrap h o f H. A componen t o f G'w having a connected star-graph St (a) wit h a t leas t on e edg e a s a subgrap h wil l b e calle d large.Suppos
e H is large. Let M be the set of all vertices a of G such that St (a) is a connecte d subgrap h o f H with at least one edge. Then IÇ V - (SUT). H is made up of these star-graphs St (a), a set N of edges which link them to for m a connecte d grap h Ho an d a se t P o f edge s havin g on e en d a verte x o f H0 an d on e en d a verte x cA such that c Ç T. Clearly M is the set of vertices of a componen t K(H) o f G SUT- We may think of K(H) as derived from H0 by shrinkin g eac h o f th e star-graph s S t (a), a G M, to a single vertex. Conversely suppos e K i s an y componen t o f G SUT- If c is a vertex of K then Y(c) ?* 0 since c $ T an d therefor e St(c ) i s connecte d an d ha s a t leas t on e edge . Thi s star - grap h i s a subgrap h o f a larg e componen t H of GV and we must have K = K(H). Fo r a larg e componen t H ol G'w having just n vertices n= Z {d(a) + (d(a)-/(a))}+»(Z(JÎ)) = £/(<*)+»(*(#) ) (mo d 2) . aeK(H)THE FACTOR THEOREM FOR FINITE GRAPHS 351
Henc e th e numbe r o f larg e component s o f G'w for which the number of vertices i s od d i s q (5, T). Usin g (8 ) w e obtai n th e formula e (9 ) hu(W) = q(S, T)+Z (d(a) - f{a)) + £ (d(c) - ds(c)), aeS ceT (10 ) hu{W) - a(W) = q(S, T) - Z/(G) + Z (f(c) - ds(c)). aeS ceT Th e quantitie s o n th e lef t i n thes e equation s ar e define d i n term s o f G' , thos e o n th e righ t i n term s o f G.Suppos
e ther e ar e disjoin t subset s S and T of V satisfying (6). Select two suc h subset s s o tha t a(S) ha s th e leas t possibl e value . Assum e that/(6 ) = d(b) fo r som e b G 5. If we replace 5 by 5 - {b} and T by IU {b} inequality (6) wil l remai n valid , fo r wit h a t mos t d(b) exception s th e number s v(C) associate d wit h th e component s o f GSuT are unaltered. This contradicts the definition of S. Henc e fib) < d(b) fo r eac h & £ 5. Let W be the union of the sets X(a) suc h tha t a £ S and the sets Y(c) such that c Ç T. Then W is simple since Y(c) i s non-nul l whe n X(c) Q W. I t follow s fro m (10 ) tha t hu(W) > a{W) i n Gf. Hence G' has no 1-factor, by Theorem A. Hence G has no/-factor, by th e Lemma .Conversel
y suppos e G has no /-factor. Then by Theorem A and the Lemma ther e i s a se t W o f vertice s o f Gr such that hu(W) > a(W). Choose such a W s o tha t a(W) ha s th e leas t possibl e value .Suppos
e ther e exist s a Ç F such that Y (a) Pi W ^ 0 and Y (a) H {V - W) ^ 0 . Writ e Z = W - ( Y (a) H W). Then G'w and G'z differ in one component only , provide d tha t X(a) i s no t a subse t o f W, sinc e th e member s o f Y (a) are al l joine d t o th e sam e vertice s o f G'. I f X{a) Q W the n eac h componen t o f G'w i s a componen t o f G'z. In either case we have hu(Z) > hu(W) - 1 and a(Z) < a(W) - 1 . Henc e hu(Z) - a{Z) > hu(W) - a(W), contrary to the definitio n o f W. W e deduc e tha t Y{a) <^W\î Y (a) C\W ^ 0.Suppos
e nex t tha t X(a) r\ W ^ 0 . Choos e b Ç X(a) Pi W. Write Z = W - {b}. Ther e i s a t mos t on e componen t o f G'w which has a vertex not a mem be r o f Y(a) joine d t o b i n G'. Hence if Y(a) is contained in W the numbers hu(Z) and hu(W) can differ by at most one. Then hu{Z) > hu{W) - 1, a(Z) = a(W) - 1 an d therefor e hu(Z) - a(Z) > hu(W) - a{W). This con tradict s th e definitio n o f W. W e deduc e that , fo r th e cas e X{a) C\W 9^ 0, Y {a) is not a subset of W and therefore Y(a) P W = 0 by the result of the precedin g paragraph . Thi s prove s tha t X(a) an d Y (a) cannot both be subsets o f W , sinc e X(a) i s neve r null . (d(a) > /(a ) > 0. )Suppos
e bot h X(a ) P P F an d X(a) P (F7 - W7) are non-null. We choose be X(a) nW an d writ e Z = I F - {b} a s before . Sinc e F(a ) P I F = 0 al l th e vertice s o f Y(a) belon g t o on e componen t o f G'w, for each is joined in G' t o eac h verte x o f X(a) P {V - W). Bu t ther e i s a t mos t on e componen t o f352 W. T. TUTTE
G1V which has a vertex not a member of Y (a) joined to b in G;. Hence with at mos t tw o exception s th e component s o f G'w are components of G''z. Accord ingl y hu(Z) > ku{W) - 2, hu(Z) - a{Z) > hu(W) - a(W) - 1. Bu t hu(Z) is by definition the number of components of G'z having an odd numbe r o f vertices . Henc e hu(Z) +a(Z) =a(V) (mod 2) an d similarl y hu(W) + a(W) s a(Vf) (mod 2). W e ma y writ e thes e result s a s hu(Z) - a{Z) s a{Vf) - hu(W) - a{W) (mod 2). Henc e hu(Z) - a(Z) > hu{W) - a(W) and so the definition of W is contra dicted . W e deduc e tha t X(a) <^ W ii X(a) C\W ^ 0 . W e hav e no w prove d tha t W i s simple . W e defin e 5 an d T i n term s o f W a s before . Usin g (10 ) w e find tha t S and T satisfy (6). Thi s complete s th e proo f o f Theore m C .REFERENCE
S 1 . H. B. Belck, Regulàre Faktoren von Graphen, J. Reine Angew. Math., 188 (1950), 228-252. 2 . D. Kônig. Théorie der endlichen und unendlichen Graphen (Leipzig, 1936). 3 . F. G. Maunsell, A note on Tutte's paper "The factorization of linear graphs,'1 J. London Math . Soc , 27
(1952) , 127-128
. 4 . W. T. Tutte, The factorization of linear graphs, J. London Math. Soc, 22 (1947), 107-111. 5 . , The factors of graphs, Can. J. Math., 4 (1952), 314-328. 6 . Hassle r Whitney , Non-separable and planar graphs, Trans . Amer. Math. Soc, 34 (1932),339-362
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