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RISS 인기검색어
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- 저자 : Minggang Zong (검색결과 2 건)
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RANDOMLY ORTHOGONAL FACTORIZATIONS OF (0,mf - (m - 1)r)-GRAPHS
Zhou, Sizhong,Zong, Minggang Korean Mathematical Society 2008 대한수학회지 Vol.45 No.6
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 $g(x)\;{\leq}\;f(x)$ for every vertex x of V(G). We use $d_G(x)$ 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 $g(x)\;{\leq}\;d_F(x)\;{\leq}\;f(x)$ 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 = {$F_1$, $F_2$, ..., $F_m$} be a factorization of G and H be a subgraph of G with mr edges. If $F_i$, $1\;{\leq}\;i\;{\leq}\;m$, has exactly r edges in common with H, we say that F is r-orthogonal to H. If for any partition {$A_1$, $A_2$, ..., $A_m$} of E(H) with $|A_i|=r$ there is a (g, f)-factorization F = {$F_1$, $F_2$, ..., $F_m$} of G such that $A_i\;{\subseteq}E(F_i)$, $1\;{\leq}\;i\;{\leq}\;m$, 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 $f(x)\;{\geq}\;3r\;-\;1$ for any $x\;{\in}\;V(G)$.
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RANDOMLY ORTHOGONAL FACTORIZATIONS OF (0,mf − (m− 1)r)-GRAPHS
Sizhong Zhou,Minggang Zong 대한수학회 2008 대한수학회지 Vol.45 No.6
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 g(x) ≤ f(x) for every vertex x of V (G). We use dG(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 g(x) ≤ dF (x) ≤ 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 = {F₁, F₂, . . . , Fm} be a factorization of G and H be a subgraph of G with mr edges. If Fi, 1 ≤ i ≤ m, has exactly r edges in common with H, we say that F is r-orthogonal to H. If for any partition {A₁,A₂, . . . ,Am} of E(H) with |Ai| = r there is a (g, f)-factorization F = {F₁, F₂, . . . , Fm} of G such that Ai ⊆ E(Fi), 1 ≤ i ≤ 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 f(x) ≥ 3r − 1 for any x ∈ V (G). 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 g(x) ≤ f(x) for every vertex x of V (G). We use dG(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 g(x) ≤ dF (x) ≤ 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 = {F₁, F₂, . . . , Fm} be a factorization of G and H be a subgraph of G with mr edges. If Fi, 1 ≤ i ≤ m, has exactly r edges in common with H, we say that F is r-orthogonal to H. If for any partition {A₁,A₂, . . . ,Am} of E(H) with |Ai| = r there is a (g, f)-factorization F = {F₁, F₂, . . . , Fm} of G such that Ai ⊆ E(Fi), 1 ≤ i ≤ 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 f(x) ≥ 3r − 1 for any x ∈ V (G).
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