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,;;;

Journal of applied mathematics & informatics, 2010. vol.28. 3_4, pp.909-912
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Randomly orthogonal factorizations in networks, Chaos, Solitons & Fractals, 2016, 93, 187
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