NEW ZEALAND JOURNAL OF MATHEMATICS Volume 29 (2000), 33-40 M INIM UM DEGREE AND F-FACTORS IN GRAPHS P. K a t e r in is a n d N. T s ik o p o u l o s (Received June 1998) Abstract. Let G be a graph, a, b two positive integers, where b > a, and suppose that 5(G) > ^ V(G) and V(G) > ^ (b + a - 3). If / is a function from V(G) into {a,a + 1,...,b} such that f ( x) is even, then G has an / - factor. 1. Introduction All graphs considered in this paper are simple and finite. Let G be a such graph. We denote by V(G) the set of vertices and E(G) the set of edges of G. If x V (G), dc(x) will denote the degree of x in G. The minimum degree of G will be denoted by 6(G). Let X and Y be two disjoint subsets of V(G). Then eo(x,y ) denotes the number and Eq (X,Y ) the set of edges of G having one end-vertex in X and the other in Y. The number of connected components of G is denoted by uj(g). For any set S of vertices in G, we define the neighbour set of S in G to be the set of all vertices adjacent to vertices in S; this set is denoted by Nq (S). Given a graph G and a function / : V(G) * N = {1,2,3,... }, we say that H is an /-factor of G if H is a spanning subgraph of G and dn(x) = f(x ) for every x e V(G). If / is the constant function taking the value k then the /-factor is said to be a fc-factor. Thus a fc-factor of G is a /c-regular spanning subgraph of G. A Hamilton cycle of G is clearly a special case of a 2-factor. A Hamilton path of G is also an /^factor, where / takes the value 1 in the two ends and the value 2 in all internal vertices. The first necessary and sufficient condition for a graph to have an /-factor was obtained by Tutte [5]. Tutte s /-Factor Theorem. A graph G has an f-factor if and only if qg(d,s ]f) + ^ 2 (f(x ) ~ dg- D(x)) < ^ 2 f ( x ) x S for all sets D, S C V(G), D f l 5 = 0 where qc(d,s-,f) denotes the number of components C of (G D) S such that ec(v (C )i S) + Ylxev(C) f ( x ) ^ - (Sometimes we refer to these as odd components.) x D 1991 AMS Mathematics Subject Classification: 05C. This work was supported financially, by the Economic Research Center of the Athens University of Economics and Business.
34 P. KATERINIS AND N. TSIKOPOULOS Tutte also proved that for any graph G and any function /, qg(d,s ;f) + '^ 2 ( f ( x ) - d G- D( x ) ) - ' ^ 2 f { x ) = f{x )(mod 2). (1 ) xes xgd x V(G) The following theorem of Dirac [1] is well known. Theorem 1. Let G be a graph such that (i> V(G ) > 3,». «D > S f» Then G has a Hamiltonian cycle. We now state a result due to Nash-Williams [4] which considerably sharpens Dirac s theorem for large graphs. Theorem 2. Let G be a graph such that S(G) > Then there exists a set of 5(Iy (G) 1 ^ (g)i+ edge-disjoint Hamiltonian cycles of G, where @ y(g) denotes 0 if ^(G) is even and 1 if 1^ ( 6?) is odd. Theorem 2 has the following corollary. Corollary 3. Let G be a graph. If 5(G) > ^V^ and V"(C?) is even, then G has 1 ^^224^~l~10^ edge-disjoint 1-factors. Proof. It follows immediately from Theorem 2 if we consider a Hamiltonian cycle as a union of two edge-disjoint 1-factors. The following theorem was proved in [3]. Theorem 4. Let G be a graph and k be a positive integer such that (i) k\v(g)\ is even, and (ii) 6(G) >, and (iii) V(G) >4/c-5. Then G has a k-factor. The main result of this paper is the following theorem, which generalizes Theorem 4. Theorem 5. Let G be a graph, a, b two positive integers, where b > a; and suppose that (i) < 5 (G )> -^ r l/(g ), a + b (ii) y(g) > ^ ( 6 + a -3 ). If f is a function from V(G) into {a, a + 1,...,b} such that Y^xev(G) f ( x ) then G has an f-factor. even, At this point we should mention the following result, which is very similar to Theorem 5 and was obtained independently by Kano and Tokushige [2].
MINIMUM DEGREE AND F-FACTORS IN GRAPHS 35 Theorem 6. Let G be a connected graph, a and b integers such that 1 < a < b and 2 < b, and let f : V(G) >{a, a + 1,..., 6} be a function such that J2xEv(G) f ( x ) is even. If F(G) > and 6(G) > then G has an f-factor. 2. Proof of Theorem 5 For the proof of Theorem 5, we shall need the following lemmas. Lemma 7. If G is Hamiltonian then for every nonempty proper subset S of V(G), oj(g S) < S. Proof. Let C be Hamiltonian cycle of G. Then for every nonempty proper subset S of V(G) we have uj(c S) < 5. But the graph C S is a spanning subgraph of G S and so uj(g S) < uj(c S). Therefore the lemma holds. Lemma 8. Let G be a graph and f a function from V(G) into N where J2x V(G) f (x) is even. Suppose that there exist D, S C V(G), D Pi S = 0 such that qg(d,s ;f) + ^2 ( f ( x ) - d G-D(x)) > ^ f ( x )- xgs x D If \D U S\ is maximal with respect to the above inequality, then (i) \Ng~d(u)\ > f(u) + 1 and \Ng(u) fl 5 < f(u) 1 for every vertex u of (G - D) - S, and (ii) V(C) > 3 for every component C of (G D) S. Proof. Let C be a component of (G D) S and let u be a vertex of C. Suppose that \Ng- d(u)\ < f(u). Define S' = S U {u}. Note that qc(d, S'; f ) > Qg(D, S; / ) 1, and Y j W * ) dg-d(x)) > x S' By the parity condition in (1), it follows that (f(x) - dg- D(x)) xss qg(d, S'] f ) + 5 3 ( / ( x) dg- D(x)) > 5 3 f(x) x(=s' which contradicts the maximality of \D U S\. Thus IA^g- d (u ) > f(u) + 1. Now suppose that \Ng(u) fl S\ > f(u). Define D' D U {w}. Then and Thus qg(d',s-,f) > qg(d,s -,f)- 1 xgd 5 3 (f(x)-d G-D'(x)) > 5 3 (f(x)-dc-d(x)) + f(u). x S x&s qg(d',s-,f) + 5 3 W * ) -d c -D '(x )) > 5 3 f(x ) x&s which again contradicts the maximality of \D U S\. Thus \Ng(u) D S\ < f(u) 1, and the proof of (i) is completed. Now (ii) follows immediately from (i) since \Ng-d(u)\ A^g (w)h5 = A^(u)n V (C ) and thus 7VG(«) n V(C)\ > 2. x D'
36 P. KATERINIS AND N. TSIKOPOULOS Proof of Theorem 5. Suppose that G does not have an /-factor. Then by Tutte s /-factor theorem, there exist D, S C V (G), D fl S = 0, such that <20 (D,S ;/) + (/(z ) -< fc _ D(z)) > / ( * ). (2) xes We assume at this point that D U S\ is maximal with respect to (2), so by using Lemma 8 (ii) we have x(ed ^ (C )! > 3 for every component C of (G D) S. (3) By using (1) and the fact that J2xev(G) f (x) eveni (2) implies qg(d,s J ) + ' ( f ( x ) - d G-.D(x)) > / ( * ) + 2. (4) x e s Note immediately that S ^ 0 ; for otherwise, since a < f(x ) < b for every x G V (G), (4) would imply qo(d, 0 ; / ) > a\d\ + 2, and so uj(g D) > a\d\ 4-2, contradicting Lemma 7 since, by conditions (i) and (ii) of Theorem 5, G is Hamiltonian. (Note that the only case where condition (ii) fails to imply V(G) > 3, is when 1 = a < b < 2; thus we may have G = in which case the conclusion of Theorem 5 holds.) Let m min{<ig-d(^) Ix G S}. We consider the following cases. Case 1. m >b + 1 Then (4) implies qdd, 5; / ) + ( 1) 5 > a\d\ + 2 since a < f { x ) < b for every x G V(G). Hence x e d qa(d,s-j) > a D + S + 2. (5) But, by conditions (i) and (ii) of the theorem, G is Hamiltonian and so, by Lemma 7, u)(g T) < T (6) for every nonempty proper subset T of V(G). Clearly (6) contradicts (5), therefore the theorem holds true in this case. Case 2. 0 < m < b 1 Then from (4) we have hence and so qg(d,s-,f) + (b-m)\s\ > a\d\ + 2, (b -m )(q G(D,S-,f) + \S\) > a\d\ + 2 qa {D,S-,}) + \S\ > (7) Since V(G) > \D\ + \S\ + qa (D,S ;f), (7) implies V(G) - D > 2 ^, and so b m * W - (8) Let u be a vertex in S such that do-d{u) = m. Then 5{G) < dg(u) <\D\ + d G- D{u) = \D\+m. (9)
MINIMUM DEGREE AND F-FACTORS IN GRAPHS 37 So, by using (i), (9) implies Combining (8) and (10) we have hence ( ^ V ( G ) - m < D. (10) Z l k > ( J L - W ) - m + 1 \a + b j 1 v b m \ / y(g) ( * m \b m + a b + a j b m + a At this point we may assume that m ^ 0 for otherwise we get a contradiction from (11). Now since, (11) may be written as SO hence V(G) < h -----m b~m+a b m _ b b m+a b+a (b + a)[2 - m{b - m + a)} ~~ (b m)(b + a) b(b m + a) V(G) < (&-m + a - - Y (12) a \ m ) Now let / be the function defined by f(m ) = 6 ra + a on {1, 2,...,b 1}. Clearly /(m ) takes its maximum value when m 1. So from (12) we have V(G) < ^ ^ ( b + a - 3). (13) a But (13) contradicts condition (ii) of the theorem; so the theorem holds in this case Case 3. m = b Then (4) implies But using (3) it follows that which along with (14) yields and since S ^ 0, we have qa(d,s-j) > a\d\ + 2. (14) 3qG(D,S\f) + D + S < V(G), (15) 3a D +6+ L> + S < \V(G)l (16), m z i. Let u be a vertex in S such that ^ g -d (w ) = m b. Then b b + a \V(G )\<6(G )<dg(u) <\D\ + b
38 P. KATERINIS AND N. TSIKOPOULOS hence b\v(g)\ a + b Combining (17) and (18) we obtain -b < \ D \. (18) Here we consider two subcases: Case 3a. b > 3 Then 3b2 126 + 10 > 0. Since b > a we have 3b2 116 a + 10 > 0, which yields 3(ib -\- b 7 < 3b2 -f- 3 106 -I- 3<z6 a = (36 1) (6 I o 3). Hence But using (20), (19) implies 3a6 ~(- 6 7 /nn\ j- < 6 + a - 3. (20) V(G) < (i> + a)(6 + tt- 3) a which contradicts condition (ii) of the theorem. Thus the theorem holds in this subcase. Case 3b. 6 < 2 If a = 6 < 2, the theorem holds trivially since G is Hamiltonian. So the only case which remains to be considered is when a = 1 and 6 = 2. But then (19) implies \V(G)\ < f, which is a contradiction. Therefore the theorem holds true in this subcase too. This completes the proof of Theorem 5. 3. Remarks We next show that conditions (i) and (ii) of Theorem 5 are in some sense the best possible. First we show this for condition (i), by constructing a family of graphs Q satisfying the following conditions for G G Q. (a) 5(G) = 6 b + a \V(G)\ - 1. (b) V(G) > (6 + a 3). a (c) There exists a function /: V(G) >{a, a + 1,..., 6} such that ^ xev(g) f ( x ) is even and G does not have an /-factor. Let y be an integer having the property that y is also an integer. Our family of graphs Q is obtained by joining every vertex of Kb%_2 to every element of a set a A containing y isolated vertices. Proposition 9. Our family of graphs Q satisfy the above conditions (a), (b), and (c).
MINIMUM DEGREE AND F-FACTORS IN GRAPHS 39 Proof. Let Gbea graph in Q. (a) Clearly 8(G) = ^ 2, and since V(G) = ^ 2 + y, we have i(g) = ±*1-2. a b So 8(G) and since 0 < < 1, we obtain a b a -f- b Now using the fact that 8(G) is an integer, we have 8(G) = 1- (b) This condition holds provided that y is sufficiently large. (c) Let / : V(G) >{a, a + 1,..., b} be a function such that f(x ) = a if x G V, and f(x ) b if x G A. It is obvious that YlxeV(G) f ( x ) = by 2a + by, which is an even number. On the other hand if we let D = V(KbJL_2) and S = A, we have qg(d,s ;f) + (f(x ) - dg-d (x)) > ^ 2 f(x ) xes xed since qg(d,s ]f) = 0, (f(x) - dg- D(x)) = 6 5 = by, and xg >/(z) = a\d\ = by 2a. Therefore, by Tutte s theorem, G has no /-factor. Finally in order to show that condition (ii) of Theorem 5 is the best possible, we construct a graph G satisfying the following conditions. (a) 5(G) > V(G). (b) V(G) = *±s(6 + a -3). (c) There exists a function / : V(G) > {a, a + 1,...,b} such that J2x v (G) f ( x ) is even and G does not have an /-factor. Let a, b be positive integers satisfying the properties: (i) a divides b, (ii) a and b have the same parity. We obtain our graph G by joining every vertex of if b(b+a_3)_1 to all the vertices of b+ ~2 copies of K 2. Proposition 10. This graph G satisfies the above conditions (a), (b) and (c). Proof. (b) Clearly F(G) = ^(b + a - 3 ) - l + b + a - 2 = ^ ( b + a - 3 ). (a) We have 8(G) (b + a 3) and since V(G) = (b + a 3), b 5(C) = b+a t h WG). (c) Let X be the set having as elements the vertices of the b+% 2 copies of K 2, and consider a function / : V(G) >{a, a + 1,..., 6} such that f(x) = a if x G K b(b+a_ 3)_1, and f(x) = b if x G X.
40 P. KATERINIS AND N. TSIKOPOULOS Clearly J2xev(G) f ( x ) = a(a(b + a - 3) - l) + 6(6 + a - 2) = 6(6 + a - 3) - a + 6(6 + a 2), and since a, 6 have the same parity, 22xev(G) f ( x ) *s an even number. On the other hand if we let D = V[Kb (b+a_3)_1) and S = X, we have qg{d,s ;f) + ^ 2 ( f ( x ) - d G- D(x)) > ^ / ( x ), xes since qg(d,s-,f) = 0, Exgd f ( x ) = 6(6 + a - 3) - a and (f(x) - dg-n {x )) = (6 1)(6 + a 2). Therefore, by Tutte s theorem, G has no /-factor. References 1. G.A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. 2 (1952), 69-81. 2. M. Kano and N. Tokushige, Binding Numbers and f-factors of Graphs, J. Combinatorial Theory, Series B, 54 (1992), 213-221. 3. P. Katerinis, Minimum degree of a graph and the existence of k-factors, Proc. Indian Acad. Sci. (Math. Sci.), 94 (1985), 123-127. 4. C. St. J.A. Nash-Williams, Edge-disjoint circuits in graphs with vertices of large valency, Stud. Pure Math., Paper presented to R. Rado, Academic Press, London and New York, 1971. 5. W.T. Tutte, The factors of graphs, Canad. J. Math. 4 (1952), 314-328. P. Katerinis and N. Tsikopoulos Department of Informatics Athens University of Economics 76 Patission Str., Athens 10434 GREECE pek@aueb.gr