MANY physical structures can conveniently be modelled

Transcription

1 Proceedings of the World Congress on Engineering Coputer Science 2017 Vol II Roly r-orthogonal (g, f)-factorizations in Networks Sizhong Zhou Abstract Let G (V (G), E(G)) be a graph, where V (G) E(G) denote its vertex set edge set, respectively. Let g, f : V (G) Z be two functions with f(x) g(x) kr for each x V (G). Let H 1, H 2,, H k be k vertex disjoint rsubgraphs of G. In this paper, it is verified that every (g ( 1)r, f ( 1)r)-graph contains a (g, f)-factorization roly r-orthogonal to each H i (1 i k). Index Ters network; graph; (g, f)-factor; r-orthogonal factorization. I. INTRODUCTION MANY physical structures can conveniently be odelled by networks. Exaples include a counication network with nodes links odelling cities counication channels, respectively, or a railroad network with nodes links representing railroad stations railways between two stations, respectively. Many probles on network design optiization, e.g., the file transfer probles on coputer networks, building blocks, coding design, scheduling probles so on, are related to the factors, factorizations orthogonal factorizations in graphs [1]. The file transfer proble can be odeled as (0, f)- factorizations (or f-colorings) in graphs [2]. The designs of Roo squares Latin squares are related to orthogonal factorizations in graphs[1]. It is well known that a network can be represented by a graph. Vertices edges of the graph correspond to nodes links between the nodes, respectively. Henceforth we use the ter graph instead of network. All graphs under consideration are finite, undirected, siple. Let G (V (G), E(G)) be a graph, where V (G) E(G) denote its vertex set edge set, respectively. For every x V (G), we use d G (x) to denote the degree of x in G. Let g, f : V (G) Z be two functions with 0 g(x) f(x) for any x V (G). A (g, f)-graph is a graph G satisfying g(x) d G (x) f(x) for arbitrary x V (G). A spanning subgraph F of a graph G is said to be a (g, f)-factor of G if F itself is a (g, f)-graph. Let r be two positive integers. A (g, f)-factorization F {F 1, F 2,, F } of a graph G is a partition of E(G) into edge-disjoint (g, f)- factors F 1, F 2,, F. A subgraph with edges is said to be an -subgraph. Let H be an r-subgraph of a graph G. A (g, f)-factorization F {F 1, F 2,, F } is said to be r-orthogonal to H if every F i (1 i ) adits exactly r edges in coon with H. A (g, f)-factorization F {F 1, F 2,, F } of G is called roly r-orthogonal to H if A i E(F i ) for arbitrary partition {A 1, A 2,, A } Manuscript received July 02, 2017; revised July 19, Sizhong Zhou is with the School of Science, Jiangsu University of Science Technology, Mengxi Road 2, Zhenjiang, Jiangsu , P. R. China e-ail: zsz cut@163.co. of E(H) satisfying A i r, 1 i. It is obvious that roly 1-orthogonal is equivalent to 1-orthogonal 1- orthogonal is also said to be orthogonal. Alspach et al. [1] put forward the following proble: For a given subgraph H of a graph G, does there exist a factorization F of a graph G with a given property orthogonal to H? Zhou et al [3], [4], [5] discussed the existence of graph factors. Research on orthogonal factorizations attracted uch attention due to their applications in cobinatorial designs, network design, circuit layout so on [1]. Feng [6] justified that every (0, f 1)-graph has a (0, f)- factorization orthogonal to any given -subgraph. Liu [7] verified that every (g 1, f 1)-graph has a (g, f)-factorization orthogonal to a atching with edges. Li Liu [8] justified that every (g 1, f 1)- graph G has a (g, f)-factorization orthogonal to any given - subgraph. La, Liu, Li Shiu [9] showed that every (g 1, f 1)-graph adits a a (g, f)-factorization orthogonal to k vertex-disjoint -subgraphs H 1, H 2,, H k. Zhou, Xu Xu [10] showed the existence of (0, f)- factorizations roly r-orthogonal to any given k vertex disjoint r-subgraphs in (0, f ( 1)r)-graphs. Liu Long [11] proved that every (g 1, f 1)- graph G has a (g, f)-factorization roly r-orthogonal to any given r-subgraph. Many authors studied the orthogonal factorizations in digraphs [12], [13], [14], [15]. In the following, we consider the ore general proble: For k given vertex-disjoint r-subgraphs H 1, H 2,, H k of G, does there exist a factorization F roly r-orthogonal to every H i (i 1, 2,, k)? We show that this proble above is true by the following theore which is our ain result in this paper. Theore 1. Let G be an (g ( 1)r, f ( 1)r)-graph, let H 1, H 2,, H k be k vertex disjoint rsubgraphs of G, where, k, r are three positive integers g, f : V (G) Z are two functions with f(x) g(x) kr for any x V (G). Then G adits a (g, f)-factorization roly r-orthogonal to each H i, 1 i k. II. PRELIMINARY LEMMAS Let G be a graph with vertex set V (G) edge set E(G). Let S be a subset of V (G) A be a subset of E(G). Then we use G S G A to denote the subgraphs induced by V (G) \ S E(G) \ A, respectively. For any function ϕ defined on V (G) X V (G), set ϕ(x) x X ϕ(x) ϕ( ) 0. For two disjoint vertex subsets S T of G, we use E G (S, T ) to denote the set of edges in G with one end in S

2 Proceedings of the World Congress on Engineering Coputer Science 2017 Vol II the other in T, write e G (S, T ) for E G (S, T ). Let S T be two disjoint vertex subsets of G, let E 1 E 2 be two disjoint edge subsets of G. Set U V (G) \ (S T ), E(S) {xy E(G) : x, y S} Define E(T ) {xy E(G) : x, y T }, E 1 E 1 E(S), E 1 E 1 E G (S, U), E 2 E 2 E(T ), E 2 E 2 E G (T, U), α(s, T ; E 1, E 2 ) 2 E 1 E 1, β(s, T ; E 1, E 2 ) 2 E 2 E 2 (S, T ; E 1, E 2 ) α(s, T ; E 1, E 2 ) β(s, T ; E 1, E 2 ). Under without abiguity, we use α, β to denote α(s, T ; E 1, E 2 ), β(s, T ; E 1, E 2 ) (S, T ; E 1, E 2 ), respectively. The following lea is very useful for proving Theore 1. Lea 2.1 (Li Liu [8]). Let G be a graph, g, f : V (G) Z be two functions defined on V (G) with 0 g(x) < f(x) d G (x) for any x V (G). Let E 1 E 2 be two disjoint edge subsets of G. Then G contains a (g, f)- factor F with E 1 E(F ) E 2 E(F ) if only if δ G (S, T ; g, f) f(s)d G S (T ) g(t ) (S, T ; E 1, E 2 ) for any two disjoint vertex subsets S T of G. In the following, we always assue that G is a (g( 1)r, f ( 1)r)-graph, where, r are two integers with 1 r 1. Define p(x) ax{g(x), d G (x) ( 1)f(x) ( 2)r} q(x) in{f(x), d G (x) ( 1)g(x) ( 2)r} for any x V (G). In view of the definitions of p(x) q(x), it is easy to see that for any x V (G). We write for any x V (G). g(x) p(x) < q(x) f(x) 1 (x) 1 d G(x) p(x) 2 (x) q(x) 1 d G(x) Lea 2.2. For any x V (G) 2, we obtain r, if p(x) > g(x) d G(x) f(x) 1 (x) ( 1)r r 1, ( 1)r, otherwise. 2 (x) r, if q(x) < f(x) d G(x) g(x) ( 1)r r 1, ( 1)r, otherwise. Proof. If p(x) g(x), then we have 1 (x) 1 d G(x) p(x) 1 d G(x) g(x) 1 (g(x) ( 1)r) g(x) ( 1)r. Otherwise, it follows fro the definition of p(x) that p(x) d G (x) ( 1)f(x) ( 2)r. If d G (x) f(x) ( 1)r r, then we obtain 1 (x) 1 d G(x) p(x) 1 d G(x) (d G (x) ( 1)f(x) ( 2)r) 1 (f(x) ( 1)r r) ( 1)f(x) ( 2)r ( 1)r r >. Otherwise, we have f(x) ( 1)r d G (x) f(x) ( 1)r r 1. Thus, we have 1 (x) 1 d G(x) p(x) 1 d G(x) (d G (x) ( 1)f(x) ( 2)r) 1 (f(x) ( 1)r) ( 1)f(x) ( 2)r r. So we prove the first inequality. Siilarly, we can verify the second inequality. This copletes the proof of Lea 2.2. Lea 2.3 (Li Liu [8]). For arbitrary two disjoint vertex subsets S T of G, the following equality holds δ G (S, T ; p, q) 1 (T ) 2 (S) 1 d G S(T ) 1 d G T (S). III. THE PROOFS OF MAIN RESULTS Let G be a graph, let g f be two integer-valued functions defined on V (G) satisfying f(x) > g(x) kr for each x V (G). Let H 1, H 2,, H k be k vertex-disjoint r-subgraphs of G. For i 1, 2,, k, we set A i1 {xy E(H i ) : p(x) g(x)1 p(y) g(y)1}, A i2 {xy E(H i ) : p(x) g(x) 1 or p(y) g(y) 1}

3 Proceedings of the World Congress on Engineering Coputer Science 2017 Vol II A i1, if A i1, A i A i2, if A i1 A i2, E(H i ), otherwise. Choose any Q i A i with Q i r for 1 i k. We write E 1 k i1 Q i E 2 ( k i1 E(H i)) \ E 1. Thus, we obtain E 1 kr E 2 ( 1)kr. For any two disjoint subsets S, T V (G), E 1, E 1, E 2, E 2, α, β are defined as in Section 2. It follows fro the definitions of α β that α in{2kr, r S } β in{2( 1)kr, ( 1)r T }. The definitions of p(x), q(x), 1 (x) 2 (x) are identical to that in Section 2. In order to prove Theore 1, we first justify the following lea which plays a crucial role for proving Theore 1. Lea 3.1. Let G be a (g ( 1)r, f ( 1)r)- graph, where 2. Then G adits a (p, q)-factor F 1 with E 1 E(F 1 ) E 2 E(F 1 ). Proof. According to Lea 2.1, it suffices to show that δ G (S, T ; p, q) α β for any two disjoint vertex subsets S T of G. If S, then α 0. In ters of Lea 2.2, Lea 2.3 the condition g(x) kr for any x V (G), we obtain δ G (S, T ; p, q) 1 (T ) 1 d G(T ) r T 1 (g(t ) ( 1)r T ) ( 1)r T β α β. If T, then β 0. In view of Lea 2.2, Lea 2.3 the condition g(x) kr for any x V (G), we have δ G (S, T ; p, q) 2 (S) 1 d G(S) r S 1 (g(s) ( 1)r S ) r S α α β. Hence, we ay assue that S T. In the following, let us consider two cases. Case 1. T k 1. Subcase 1.1. S k 1. According to Lea 2.2, Lea 2.3 the condition g(x) kr for any x V (G), we obtain δ G (S, T ; p, q) 1 d G S(T ) 1 d G T (S) 1 (g(t ) ( 1)r T S T ) 1 (g(s) ( 1)r S T S ) 1 (kr ( 1)r S ) T 1 (kr ( 1)r T ) S 1 (kr ( 1)r k 1) T 1 (kr ( 1)r k 1) S 1 (kr k ( 1)r 1) T 1 (kr k ( 1)r 1) S 1 (r ( 1) 1) T 1 (r ( 1) 1) S ( 1)r T r S α β. Subcase 1.2. S k. Note that d G S (T ) β. It follows fro Lea 2.2, Lea 2.3, 2 the condition g(x) kr for any x V (G) that δ G (S, T ; p, q) 2 (S) 1 d G S(T ) 1 d G T (S) r S ( 1)β 1 (g(s) ( 1)r S T S ) r S ( 1)β 1 (kr ( 1)r T ) S r S ( 1)β 1 (kr ( 1)r k 1) S ( 1)β kr 1 (kr k ( 1)r 1)k ( 1)β kr (r ( 1) 1)k ( 1)β 2kr ( 1)kr ( 1)β 2( 1)kr α ( 1)β α β α β. Case 2. T k. Note that d G T (S) α d G S (T ) β. Now we choose v T with d G[E2](v) in{d G[E2](x) : x T }. Cobining this with T k E 2 ( 1)kr, we

4 Proceedings of the World Congress on Engineering Coputer Science 2017 Vol II have S d G S (T ) S d G S (v) d G S (T \ {v}) 2( 1)kr d G (v) β k 2( 1)r d G (v) β (1) Observe that for soe u S. Set T d G T (S) d G (u) α r (2) S 0 {x S : q(x) f(x) or d G (x) g(x)( 1)rr}, S 1 S \ S 0, T 0 {x T : p(x) g(x) or d G (x) f(x) ( 1)r r}, Furtherore, we set T 1 T \ T 0. S 0 {x S 0 : p(x) g(x) 1}, S 1 S \ S 0, T 0 {x T 0 : p(x) g(x)}, T 1 T \ T 0. It is obvious that p(x) g(x) for each x S 1 T 0 p(x) g(x) 1 for each x S 0 T 1. Subcase 2.1. S 0 T 0 2(k 1). Note that S T. According to Lea 2.2, Lea 2.3, 2, (1), (2) the condition g(x) kr for any x V (G), we obtain δ G (S, T ; p, q) 1 (T ) 2 (S) 1 d G S(T ) 1 d G T (S) r T 1 ( 1)r T 0 r S 1 ( 1)r S 0 1 d G S(T ) 1 d G T (S) r T ( 2)r T 0 r S ( 2)r S 0 1 d G S(T ) 1 d G T (S) (r 1) S (r 1) T ( 2)r( S 0 T 0 ) T d G T (S) S d G S(T ) 2 d G S(T ) 2(r 1) 2( 2)(k 1)r d G(u) α r d G(v) β 2( 1)r ( 2)β 2( 2)(k 1)r g(u) > ( 1)r α r g(v) ( 1)r ( 1)β 2 2( 2)kr 2kr α ( 1)β r 2 2( 1)kr 2( 1)kr α ( 1)β r 2 ( 1)α β α ( 1)β 1 > α β 1. Since δ G (S, T ; p, q) is an integer, we obtain δ G (S, T ; p, q) α β. Subcase 2.2. S 0 T 0 < 2(k 1). We write n V (E 1 ) T 1. Then we obtain V (E 1 ) (S 0 T 1) S 0 n. (3) Let Q i E 1 E(H i ), where V (Q i ) S T, Q i r V (Q i ) q i (2 q i 2r). According to the choice of E 1, if V (Q i ) (S 0 T 1), then V (H i ) T 1 ; if V (Q i ) (S 0 T 1) 1, then V (Q) T 1 1 for any Q E(H i ) with Q r V (Q) q i ; if V (Q i ) (S 0 T 1) 2, then V (Q) T 1 2 for any Q E(H i ) with Q r V (Q) q i ;...; if V (Q i ) (S 0 T 1) q i, then V (Q) T 1 q i for any Q E(H i ) with Q r V (Q) q i. We write n V (E 1 ) U, where U V (G) \ (S T ). Cobining these with (3) the definition of β(s, T ), we have β(s, T 1) V (E 1 ) (S 0 T 1) ( 1)r n ( 1)r Hence, we obtain ( S 0 n n )( 1)r. β(s, T ) β(s, T 0) β(s, T 1) T 0 ( 1)r ( S 0 n n )( 1)r ( S 0 T 0 n n )( 1)r, (4) α(s, T ) 2kr (n n )r. (5) Note that S T. In ters of Lea 2.2, Lea 2.3, 2, (1), (2), (4), (5) the condition g(x)

5 Proceedings of the World Congress on Engineering Coputer Science 2017 Vol II kr for any x V (G), we obtain δ G (S, T ; p, q) 1 (T ) 2 (S) 1 d G S(T ) 1 d G T (S) r T 1 ( 1)r T 0 r S 1 ( 1)r S 0 1 d G S(T ) 1 d G T (S) r T ( 2)r T 0 r S ( 2)r S 0 1 d G S(T ) 1 d G T (S) ( 2)r( S 0 T 0 ) S d G S(T ) ( 2)r( S 0 T 0 ) d G(v) β 2( 1)r ( 2)r( S 0 T 0 ) T d G T (S) 2 d G S(T ) d G(u) α r ( 2)β g(u) ( 1)r α r g(v) ( 1)r ( 1)β 2( 1)r 2 ( 2)r( S 0 T 0 ) 2kr 2( 1)r α ( 1)β r 2( 1)r 2 > ( 2)r( S 0 T 0 ) 2kr α ( 1)β 2r ( 2)r( S 0 T 0 ) 2(k 1)r 2( 1)kr α ( 1)β > ( 1)r( S 0 T 0 ) ( 1)(α (n n )r) α ( 1)β ( 1)r( S 0 T 0 n n ) α ( 1)β α β α β. This copletes the proof of Lea 3.1. Proof of Theore 1. We apply induction on. Theore 1 holds for 1. We assue that Theore 1 is true for 1, where 2. In view of Lea 3.1, G adits a (p, q)-factor F 1 with E 1 E(F 1 ) E 2 E(F 1 ). It is obvious that F 1 is also a (g, f)-factor of G. We write G G E(F 1 ). In ters of the definitions of p(x) q(x), we have d G (x) d G (x) d F1 (x) d G (x) q(x) d G (x) (d G (x) ( 1)g(x) ( 2)r) ( 1)g(x) ( 2)r. Siilarly, we obtain d G (x) d G (x) d F1 (x) d G (x) p(x) ( 1)f(x) ( 2)r. Therefore, G is an (( 1)g ( 2)r, ( 1)f ( 2)r)-graph. Set H i H i E 1, i 1, 2,, k. According to the induction hypothesis, G has a (g, f)- factorization F {F 2,, F } roly r-orthogonal to each H i, 1 i k. Thus, G has a (g, f)-factorization F {F 1, F 2,, F } roly r-orthogonal to each H i, i 1, 2,, k. This copletes the proof of Theore 1. ACKNOWLEDGMENT The author would like to express his deep gratefulness to the anonyous referees for his helpful coents valuable suggestions. REFERENCES [1] B. Alspach, K. Heinrich, G. Liu, Conteporary Design Theory A Collection of Surveys, John Wiley Sons, New York, 1992, pp [2] X. Zhou, T. Nishizeki, Edge-coloring f-coloring for Vatious Classes of Graphs, Lecture Notes in Coputer Science, vol. 834, pp , [3] S. Zhou, Soe results about coponent factors in graphs, RAIRO- Operations Research, doi: /ro/ [4] S. Zhou, J. Wu, T. Zhang, The existence of P 3 -factor covered graphs, Discussiones Matheaticae Graph Theory, doi: /dgt [5] S. Zhou, F. Yang, Z. Sun, A neighborhood condition for fractional ID- [a, b]-factor-critical graphs, Discussiones Matheaticae Graph Theory, vol. 36, no. 2, pp , [6] H. Feng, On orthogonal (0, f)-factorizations, Acta Mateatica Scientia, vol. 19, pp , [7] G. Liu, Orthogonal (g, f)-factorizations in graphs, Discrete Matheatics, vol. 143, pp , [8] G. Li, G. Liu, (g, f)-factorizations Orthogonal to a Subgraph in Graphs, Science in China (Series A), vol. 49, pp , [9] P. C. B. La, G. Liu, G. Li, W. Shiu, Orthogonal (g, f)-factorizations in networks, Networks, vol. 35, pp , [10] S. Zhou, L. Xu, Y. Xu, Roly orthogonal factorizations in networks, Chaos, Solitons & Fractals, vol. 93, pp , [11] G. Liu, H. Long, Roly orthogonal (g, f)-factorizations in graphs, Acta Matheaticae Applicatae Sinica, English Series, vol. 18, pp , [12] C. Wang, Subdigraphs with orthogonal factorizations of digraphs, European Journal of Cobinatorics, vol. 33, pp , [13] S. Zhou, Q. Bian, Subdigraphs with orthogonal factorizations of digraphs (II), European Journal of Cobinatorics, vol. 36, pp , [14] S. Zhou, Rearks on orthogonal factorizations of digraphs, International Journal of Coputer Matheatics, vol. 91, pp , [15] S. Zhou, Z. Sun, Z. Xu, A result on r-orthogonal factorizations in digraphs, European Journal of Cobinatorics, vol. 65, pp , 2017.