Orthogonal (g, f)-factorizations in networks

Abstract

Let G = (V, E) be a graph and let g and f be two integer-valued functions defined on V, k and shown that there are polynomial-time algorithms to find the desired (g, f )-factorizations.

On k-orthogonal factorizations in networks

“…The following lemma, whose proof can be shown by Lam et al [7], Li et al [10] and Liu et al [9], is useful for verifying our main theorem. Lemma 2.1 ( [7,9,10]). Let G be a graph, and let g, f :…”

“…Li and Liu [9] verified that every (mg + m − 1, mf − m + 1)graph admits a (g, f )-factorization orthogonal to any given m-subgraph, which is an improvement of Liu's previous result [11]. Lam et al [7] justified that every (mg + m − 1, mf − m + 1)-graph admits a (g, f )factorization orthogonal to k vertex-disjoint m-subgraphs if k ≤ g(x) ≤ f (x) for any x ∈ V (G), which is a generalization of Li and Liu's previous result [9]. Li et al [10] investigated the existence of a subgraph with orthogonal factorization in an (mg + k, mf − k)-graph.…”

On k-orthogonal factorizations in networks

“…The following lemma, whose proof can be shown by Lam et al [7], Li et al [10] and Liu et al [9], is useful for verifying our main theorem. Lemma 2.1 ( [7,9,10]). Let G be a graph, and let g, f :…”

“…Li and Liu [9] verified that every (mg + m − 1, mf − m + 1)graph admits a (g, f )-factorization orthogonal to any given m-subgraph, which is an improvement of Liu's previous result [11]. Lam et al [7] justified that every (mg + m − 1, mf − m + 1)-graph admits a (g, f )factorization orthogonal to k vertex-disjoint m-subgraphs if k ≤ g(x) ≤ f (x) for any x ∈ V (G), which is a generalization of Li and Liu's previous result [9]. Li et al [10] investigated the existence of a subgraph with orthogonal factorization in an (mg + k, mf − k)-graph.…”

Toughness and [a, b]-factors with inclusion/exclusion properties

“…Then β(S, T ; E 1 , E 2 ) = 2|E 2 ∩ E G (T )| + |E 2 ∩ E G (T, U )| = 0. It follows from(7)and(8) again that α(S, T ; E 1 , E 2 ) = 0. Thus, |E 1 | 1, and then E 1 = {e 1 } or E 1 = {e 1 , e 2 }.…”

Orthogonal factorizations of digraphs

“…For an undirected graph, Alspach [2] posed the following problem: given a subgraph H, does there exist a factorization F of G with some fixed type orthogonal to H? Liu et al [4,[7][8][9][10] discussed the orthogonal (g, f )-factorizations of graphs. Kano [6] studied the [a, b]-factorizations in graphs.…”

On k-orthogonal factorizations in networks

“…The following lemma, whose proof can be shown by Lam et al [7], Li et al [10] and Liu et al [9], is useful for verifying our main theorem. Lemma 2.1 ( [7,9,10]). Let G be a graph, and let g, f :…”

“…Li and Liu [9] verified that every (mg + m − 1, mf − m + 1)graph admits a (g, f )-factorization orthogonal to any given m-subgraph, which is an improvement of Liu's previous result [11]. Lam et al [7] justified that every (mg + m − 1, mf − m + 1)-graph admits a (g, f )factorization orthogonal to k vertex-disjoint m-subgraphs if k ≤ g(x) ≤ f (x) for any x ∈ V (G), which is a generalization of Li and Liu's previous result [9]. Li et al [10] investigated the existence of a subgraph with orthogonal factorization in an (mg + k, mf − k)-graph.…”

Toughness and [a, b]-factors with inclusion/exclusion properties

“…Then β(S, T ; E 1 , E 2 ) = 2|E 2 ∩ E G (T )| + |E 2 ∩ E G (T, U )| = 0. It follows from(7)and(8) again that α(S, T ; E 1 , E 2 ) = 0. Thus, |E 1 | 1, and then E 1 = {e 1 } or E 1 = {e 1 , e 2 }.…”

Orthogonal factorizations of digraphs

“…For an undirected graph, Alspach [2] posed the following problem: given a subgraph H, does there exist a factorization F of G with some fixed type orthogonal to H? Liu et al [4,[7][8][9][10] discussed the orthogonal (g, f )-factorizations of graphs. Kano [6] studied the [a, b]-factorizations in graphs.…”

On k-orthogonal factorizations in networks

“…The following lemma, whose proof can be shown by Lam et al [7], Li et al [10] and Liu et al [9], is useful for verifying our main theorem. Lemma 2.1 ( [7,9,10]). Let G be a graph, and let g, f :…”

“…Li and Liu [9] verified that every (mg + m − 1, mf − m + 1)graph admits a (g, f )-factorization orthogonal to any given m-subgraph, which is an improvement of Liu's previous result [11]. Lam et al [7] justified that every (mg + m − 1, mf − m + 1)-graph admits a (g, f )factorization orthogonal to k vertex-disjoint m-subgraphs if k ≤ g(x) ≤ f (x) for any x ∈ V (G), which is a generalization of Li and Liu's previous result [9]. Li et al [10] investigated the existence of a subgraph with orthogonal factorization in an (mg + k, mf − k)-graph.…”

Toughness and [a, b]-factors with inclusion/exclusion properties

“…Then β(S, T ; E 1 , E 2 ) = 2|E 2 ∩ E G (T )| + |E 2 ∩ E G (T, U )| = 0. It follows from(7)and(8) again that α(S, T ; E 1 , E 2 ) = 0. Thus, |E 1 | 1, and then E 1 = {e 1 } or E 1 = {e 1 , e 2 }.…”

Orthogonal factorizations of digraphs

“…For an undirected graph, Alspach [2] posed the following problem: given a subgraph H, does there exist a factorization F of G with some fixed type orthogonal to H? Liu et al [4,[7][8][9][10] discussed the orthogonal (g, f )-factorizations of graphs. Kano [6] studied the [a, b]-factorizations in graphs.…”