RANDOMLY ORTHOGONAL FACTORIZATIONS OF (0,mf - (m - 1)r)-GRAPHS

Title & Authors
RANDOMLY ORTHOGONAL FACTORIZATIONS OF (0,mf - (m - 1)r)-GRAPHS
Zhou, Sizhong; Zong, Minggang;

Abstract
Let G be a graph with vertex set V(G) and edge set E(G), and let g, f be two nonnegative integer-valued functions defined on V(G) such that $\small{g(x)\;{\leq}\;f(x)}$ for every vertex x of V(G). We use $\small{d_G(x)}$ to denote the degree of a vertex x of G. A (g, f)-factor of G is a spanning subgraph F of G such that $\small{g(x)\;{\leq}\;d_F(x)\;{\leq}\;f(x)}$ for every vertex x of V(F). In particular, G is called a (g, f)-graph if G itself is a (g, f)-factor. A (g, f)-factorization of G is a partition of E(G) into edge-disjoint (g, f)-factors. Let F = {$\small{F_1}$, $\small{F_2}$, ..., $\small{F_m}$} be a factorization of G and H be a subgraph of G with mr edges. If $\small{F_i}$, $\small{1\;{\leq}\;i\;{\leq}\;m}$, has exactly r edges in common with H, we say that F is r-orthogonal to H. If for any partition {$\small{A_1}$, $\small{A_2}$, ..., $\small{A_m}$} of E(H) with $\small{|A_i|=r}$ there is a (g, f)-factorization F = {$\small{F_1}$, $\small{F_2}$, ..., $\small{F_m}$} of G such that $\small{A_i\;{\subseteq}E(F_i)}$, $\small{1\;{\leq}\;i\;{\leq}\;m}$, then we say that G has (g, f)-factorizations randomly r-orthogonal to H. In this paper it is proved that every (0, mf - (m - 1)r)-graph has (0, f)-factorizations randomly r-orthogonal to any given subgraph with mr edges if $\small{f(x)\;{\geq}\;3r\;-\;1}$ for any $\small{x\;{\in}\;V(G)}$.
Keywords
graph;subgraph;factor;orthogonal factorization;
Language
English
Cited by
1.
NOTE ON CONNECTED (g, f)-FACTORS OF GRAPHS,Zhou, Sizhong;Wu, Jiancheng;Pan, Quanru;

Journal of applied mathematics & informatics, 2010. vol.28. 3_4, pp.909-912
References
1.
J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier Publishing Co., Inc., New York, 1976

2.
H. Feng, On orthogonal (0, f)-factorizations, Acta Math. Sci. (English Ed.) 19 (1999), no. 3, 332-336

3.
M. Kano, [a, b]-factorization of a graph, J. Graph Theory 9 (1985), no. 1, 129-146

4.
G. Li and G. Liu, (g, f)-factorization orthogonal to any subgraph, Sci. China Ser.A 27 (1997), no. 12, 1083-1088

5.
G. Liu, (g, f)-factorizations orthogonal to a star in graphs, Sci. China Ser. A 38 (1995), no. 7, 805-812

6.
G. Liu, Orthogonal (g, f)-factorizations in graphs, Discrete Math. 143 (1995), no. 1-3, 153-158

7.
G. Liu and B. Zhu, Some problems on factorizations with constraints in bipartite graphs, Discrete Appl. Math. 128 (2003), no. 2-3, 421-434

8.
L. Lovasz, Subgraphs with prescribed valencies, J. Combinatorial Theory 8 (1970), 391-416

9.
T. Tokuda, Connected [a, b]-factors in $K_{1.n}$-free graphs containing an [a, b]-factor, Discrete Math. 207 (1999), no. 1-3, 293-298

10.
J. Yuan and J. Yu, Random (m, $\gamma$)-orthogonal (g, f)-factorizable graphs, Appl. Math. A J. Chinese Univ. (Ser. A) 13 (1998), no. 3, 311-318