Binding Numbers for all Fractional (a, b, k)-critical Graphs

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1 Filomt 28:4 (2014), DOI /FIL Z Published by Fculty of Sciences nd Mthemtics, University of Niš, Serbi Avilble t: Binding Numbers for ll Frctionl (, b, k)-criticl Grphs Sizhong Zhou, Qiuxing Bin, Zhiren Sun b School of Mthemtics nd Physics, Jingsu University of Science nd Technology, Mengxi Rod 2, Zhenjing, Jingsu , P. R. Chin b School of Mthemticl Sciences, Nnjing Norml University, Nnjing, Jingsu , P. R. Chin Abstrct. Let G be grph of order n, nd let, b nd k nonnegtive integers with 2 b. A grph G is clled ll frctionl (, b, k)-criticl if fter deleting ny k vertices of G the remining grph of G hs ll frctionl [, b]-fctors. In this pper, it is proved tht G is ll frctionl (, b, k)-criticl if ()( + b 3) + n + k ()(n 1) nd bind(g) >. Furthermore, it is shown tht this 1 n k ( + b) + 2 result is best possible in some sense. 1. Introduction We consider finite undirected grphs without loops or multiple edges. Let G be grph with vertex set V(G) nd n edge set E(G). For x V(G), the set of vertices djcent to x in G is sid to be the neighborhood of x, denoted by N G (x). For ny X V(G), we write N G (X) x X N G (x). For two disjoint subsets S nd T of V(G), we denote by e G (S, T) the number of edges with one end in S nd the other end in T. Thus e G (x, V(G) \ {x}) d G (x) is the degree of x nd δ(g) min{d G (x) : x V(G)} is the minimum degree of G. For S V(G), we use G[S] to denote the subgrph of G induced by S, nd G S to denote the subgrph obtined from G by deleting vertices in S together with the edges incident to vertices in S. A vertex set S V(G) is clled independent if G[S] hs no edges. The binding number of G is defined s bind(g) min{ N G(X) X : X V(G), N G (X) V(G)}. Let nd f be two integer-vlued functions defined on V(G) with 0 (x) f (x) for ech x V(G). A (, f )-fctor of grph G is defined s spnning subgrph F of G such tht (x) d F (x) f (x) for ech x V(G). We sy tht G hs ll (, f )-fctors if G hs n r-fctor for every r : V(G) Z + such tht (x) r(x) f (x) for ech x V(G) nd r(v(g)) is even. A frctionl (, f )-indictor function is function h tht ssigns to ech edge of grph G rel number in the intervl [0,1] so tht for ech vertex x we hve (x) h(e x ) f (x), where E x {e : e xy E(G)} nd h(e x ) e E x h(e). Let h be frctionl (, f )-indictor function of grph G. Set E h {e : e E(G), h(e) > 0}. If G h is spnning subgrph of G such tht E(G h ) E h, then G h is clled frctionl (, f )-fctor of G. h 2010 Mthemtics Subject Clssifiction. Primry 05C70; Secondry 05C72, 05C35 Keywords. grph, binding number, frctionl [, b]-fctor, ll frctionl [, b]-fctors, ll frctionl (, b, k)-criticl. Received: 25 June 2013; Accepted: 08 September 2013 Communicted by Frncesco Belrdo Reserch supported by the Ntionl Nturl Science Foundtion of Chin (Grnt No ) Emil ddresses: zsz_cumt@163.com (Sizhong Zhou), binqx_1@163.com (Qiuxing Bin), 05119@njnu.edu.cn (Zhiren Sun)

2 S. Zhou et l. / Filomt 28:4 (2014), is lso clled the indictor function of G h. If h(e) {0, 1} for every e, then G h is just (, f )-fctor of G. A frctionl (, f )-fctor is frctionl f -fctor if (x) f (x) for ech x V(G). A frctionl (, f )-fctor is frctionl [, b]-fctor if (x) nd f (x) b for ech x V(G). We sy tht G hs ll frctionl (, f )-fctors if G hs frctionl r-fctor for every r : V(G) Z + such tht (x) r(x) f (x) for ech x V(G). All frctionl (, f )-fctors re sid to be ll frctionl [, b]-fctors if (x) nd f (x) b for ech x V(G). A grph G is ll frctionl (, b, k)-criticl if fter deleting ny k vertices of G the remining grph of G hs ll frctionl [, b]-fctors. Mny uthors hve investigted fctors [1,2,8] nd frctionl fctors [3,4,7,10] of grphs. The following results on ll (, f )-fctors, ll frctionl [, b]-fctors nd ll frctionl (, b, k)-criticl grphs re known. Theorem 1.1. (Niessen [6]). G hs ll (, f )-fctors if nd only if { 1, i f f (S) + d G S (x) f (T) h G (S, T,, f ) 0, i f f for ll disjoint subsets S, T V(G), where h G (S, T,, f ) denotes the number of components C of G (S T) such tht there exists vertex v V(C) with (v) < f (v) or e G (V(C), T) + f (V(C)) 1 (mod 2). 2( + b)() Theorem 1.2. (Lu [5]). Let b be two positive integers. Let G be grph with order n nd minimum degree δ(g) ()2 + 4b. If N G (x) N G (y) bn for ny two nondjcent vertices x nd y in 4 + b G, then G hs ll frctionl [, b]-fctors. Theorem 1.3. (Zhou [9]). Let, b nd k be nonnegtive integers with 1 b, nd let G be grph of order n with n + k + 1. Then G is ll frctionl (, b, k)-criticl if nd only if for ny S V(G) with S k S + d G S (x) b T k, where T {x : x V(G) \ S, d G S (x) < b}. Using Theorem 3, Zhou [9] obtined neighborhood condition for grphs to be ll frctionl (, b, k)- criticl grphs. Theorem 1.4. (Zhou [9]). Let, b, k, r be nonnegtive integers with 1 b nd r 2. Let G be grph of order ( + b)(r( + b) 2) + k (r 1)b2 bn + k n with n >. If δ(g) + k, nd N G (x 1 ) N G (x 2 ) N G (x r ) + b for ny independent subset {x 1, x 2,, x r } in G, then G is ll frctionl (, b, k)-criticl. 2. Min Result nd Its Proof In this pper, we proceed to study the existence of ll frctionl (, b, k)-criticl grphs nd obtin binding number condition for grphs to be ll frctionl (, b, k)-criticl. Our min result is the following theorem. Theorem 2.1. Let, b nd k be nonnegtive integers with 2 b, nd let G be grph of order n with ()( + b 3) + n + k ()(n 1). If bind(g) >, then G is ll frctionl (, b, k)-criticl. 1 n k ( + b) + 2 Proof. Suppose tht G stisfies the ssumption of Theorem 2.1, but it is not ll frctionl (, b, k)-criticl. Then by Theorem 1.3, there exists some subset S of V(G) with S k such tht S + d G S (x) b T k 1, (1)

3 S. Zhou et l. / Filomt 28:4 (2014), where T {x : x V(G) \ S, d G S (x) < b}. Clerly, T by (1). Define h min{d G S (x) : x T}. In terms of the definition of T, we obtin 0 h b 1. Now in order to prove the correctness of Theorem 2.1, we shll deduce some contrdictions ccording to the following two cses. Cse 1. h 0. Let X {x : x T, d G S (x) 0}. Obviously, X nd N G (V(G)\S) X, nd so N G (V(G)\S) n X. According to the definition of bind(g) nd the condition of Theorem 2.1, we hve which implies ()(n 1) n k ( + b) + 2 < bind(g) N G(V(G) \ S) n X V(G) \ S n S, ()(n 1) S > ()(n 1)n (n k ( + b) + 2)n + (n k ( + b) + 2) X Thus, we obtin Using (1), (2) nd S + T n, we hve k 1 S + d G S (x) b T (b 1)(n 1)n + (b 2)n + kn + (n k ( + b) + 2) X (b 1)(n 1)n + kn + (n k ( + b) + 2) X (b 1)(n 1)n + kn + [(n 1) + ( 1)n k ( + b) + 3] X ( ()( + b 3) + (b 1)(n 1)n + kn + [(n 1) + ( 1) k ( + b) + 3] X > (b 1)(n 1)n + kn + [(n 1) + ( 1)( + b 3) ( + b) + 3] X S + T X b T S (b 1) T X S (b 1)(n S ) X (b 1)(n 1)n + k(n 1) + (n 1) X. S > () S (b 1)n X (b 1)n + k + X > () (b 1)n X k, + k ) 1 (b 1)n + k + X. (2) which is contrdiction. Cse 2. 1 h b 1. (b 1)n + k + + b 2 Clim 1. δ(g) >. Let v be vertex of G with degree δ(g). Set Y V(G) \ N G (v). Obviously, Y nd v N G (Y). In terms of the definition of bind(g), we hve which implies ()(n 1) n k ( + b) + 2 < bind(g) N G(Y) n 1 Y n δ(g), δ(g) > (b 1)n + k + + b 2.

4 This completes the proof of Clim 1. Note tht δ(g) S + h. Then using Clim 1, we hve S. Zhou et l. / Filomt 28:4 (2014), S δ(g) h > (b 1)n + k + + b 2 h. (3) n k ( + b) + 1 Clim 2. T + h. n k ( + b) + 2 Assume tht T +h. We choose u T such tht d G S (u) h nd let Y T \N G S (u). Note tht N G S (u) d G S (u) h. Thus, we obtin nd Y T d G S (u) n k ( + b) + 2 T h > 0 N G (Y) V(G). Combining these with the definition of bind(g), we hve bind(g) N G(Y) Y n 1 ()(n 1) T h n k ( + b) + 2, which contrdicts tht the condition of Theorem 2.1. The proof of Clim 2 is completed. According to (1), (3) nd Clim 2, we obtin k 1 S + d G S (x) b T S (b h) T tht is, ( ) ( ) (b 1)n + k + + b 2 n k ( + b) + 1 > h (b h) + h (h 1)n + ( + b h)k (h 1)( + b h), k 1 > (h 1)n + ( + b h)k (h 1)( + b h). (4) (h 1)n + ( + b h)k Let f (h) (h 1)( + b h). If h 1, then by (4) we hve k 1 > f (h) f (1) k > k 1, which is contrdiction. In the following, we ssume tht 2 h b 1. In view of 2 h b 1 nd n ()( + b 3) + f (h) n k ( + b h) + (h 1) n k 2h + ( + b + 1) ()( + b 3) ( + b + 1) > 0. Thus, we obtin + k, we hve 1 f (h) f (2). (5)

5 From (4), (5) nd n S. Zhou et l. / Filomt 28:4 (2014), ()( + b 3) + + k, we obtin 1 n + ( + b 2)k k 1 > f (h) f (2) ( + b 2) ()( + b 3) + + k + ( + b 2)k ( + b 2) k 1, which is contrdiction. This completes the proof of Theorem 2.1. Remrk. In Theorem 2.1, the lower bound on the condition bind(g) is best possible in the sense since ()(n 1) ()(n 1) we cnnot replce bind(g) > with bind(g), which is shown in the n k ( + b) + 2 n k ( + b) + 2 following exmple. (b 1) + b(b 2) + ()k Let b 2, k 0 be three integers such tht + b + k 1 is even nd is positive integer. Set l + b + k 1 (b 1) + b(b 2) + ()k nd m. We construct grph 2 (b 1) + b(b 2) + ()k G K m K 2l. Then n m + 2l + + b + k 1. Let X V(lK 2 ), for ny x X, then N G (X \ x) n 1. According to the definition of bind(g), we obtin bind(g) N G(X \ x) X \ x n 1 2l 1 n 1 ()(n 1) + b + k 2 n k ( + b) + 2. Let S V(K m ), T V(lK 2 ). Then S m k, T 2l nd d G S (x) 2l. Thus, we hve S + d G S (x) b T m 2l(b 1) (b 1) + b(b 2) + ()k (b 1)( + b + k 1) k 1 < k. In terms of Theorem 1.3, G is not ll frctionl (, b, k)-criticl. Acknowledgments. The uthors re grteful to the nonymous referee for his vluble suggestions for improvements of the presenttion. References [1] C. Chen, Binding number nd minimum degree for [, b]-fctor, Journl of Systems Science nd Mthemticl Sciences (Chin) 6(1) (1993) [2] J. R. Corre, M. Mtml, Some remrks bout fctors of grphs, Journl of Grph Theory 57 (2008) [3] G. Liu, Q. Yu, L. Zhng, Mximum frctionl fctors in grphs, Applied Mthemtics Letters 20(12) (2007) [4] G. Liu, L. Zhng, Toughness nd the existence of frctionl k-fctors of grphs, Discrete Mthemtics 308 (2008) [5] H. Lu, Simplified existence theorems on ll frctionl [, b]-fctors, Discrete Applied Mthemtics 161(13 14) (2013) [6] T. Niessen, A chrcteriztion of grphs hving ll (, f )-fctors, Journl of Combintoril Theory Series B 72 (1998) [7] S. Zhou, A sufficient condition for grphs to be frctionl (k, m)-deleted grphs, Applied Mthemtics Letters 24(9) (2011) [8] S. Zhou, Independence number, connectivity nd (, b, k)-criticl grphs, Discrete Mthemtics 309(12) (2009) [9] S. Zhou, Z. Sun, On ll frctionl (, b, k)-criticl grphs, Act Mthemtic Sinic, English Series 30(4) (2014) [10] S. Zhou, L. Xu, Z. Sun, Independence number nd minimum degree for frctionl ID-k-fctor-criticl grphs, Aequtiones Mthemtice 84(1 2) (2012)