Randomly Orthogonal (p, f)-factorizations in Graphs

Let G be a graph with vertex set V(G) and edge set E(G) and let g and f be two integer-valued functions defined on V(G) such that 2k-1≤g(x) ≤ f(x) for all x ∈ V(G). Let H be a subgraph of G with mk edges . In this paper it is proved that every (mg m - 1,mf- m 1)-graph G has (g, f)-factorizations randomly κ-orthogonal to H and shown that the result is best possible.     

ORTHOGONAL (g,f)-FACTORIZAFIONS OF BIPARTITE GRAPHS

1 IntroductionIn this paper we con8ider finite undirected simple graphs. Let G be a graph with vertexset V(G) and edge set E(G). Let g and f be two po8itive iuteger-valued functions defined onV(G) such that g(x) 5 f(x) for every vertex x of V(G). Then a (g, f)-factor of G is a spanningsubgraph H of G satisfying g(x) 5 dH(x) 5 f(x) for each x E V(H). In particular, if G itselfis a (g, f)-factor, then G is called a (g, f)-grapl1. A subgrapl1 H of G is called an rmsubgraphif H has m edg…     

(g,f)-FACTORS WITH SPECIAL PROPERTIES IN BIPARTITE (mg,mf)-GRAPHS

§1　Preliminary and resultsAll graphs considered in this paper are finite graphs which may have multiple edgesbut no loops.Let G be a graph with vertex set V( G) and edge set E( G) .The degree of avertex x is denoted by d G( x) .The connectivity and edge-connectivity of G are denoted byκ( G) andλ( G) ,respectively.Letg and f be two positive integer-valued functions definedon vertex set V( G) such that g ( x)≤f ( x) .Then a ( g,f ) -factor of G is a spanningsubgraph H of G satisfying…     

A polynomial algorithm for finding (g,f)-colorings orthogonal to stars in bipartite graphs

Let G be a bipartite graph with vertex set V(G) and edge set E(G), and let g and f be two nonnegative integer-valued functions defined on V(G) such that g(x)≤ f(x) for every vertex x of V(G). A (g. f)-coloring of G is a generalized edge-coloring in which each color appears at each vertex x at least g(x) and at most f(x) times. In this paper a polynomial algorithm to find a (g. f)-coloring of a bipartite graph with some constraints using the minimum number of colors is given. Furthermore, we show that the results in this paper are best possible.     

FRACTIONAL (g, f)-FACTORS OF GRAPHS

1 IntroductionThe graphs considered in this paper will be finite undirected graphs wllicll 11lay llavemultiple edges but no loops. Let G be a grapll with vertex set V(G) and edge set E(G). Fora vertex x of G, the degree of x in G is denoted by dG(z). Let g and f be two integer-valuedfunctions defined o11 V(G) such that 0 < g(z) 5 f(x) fOr all x E V(G). Then a (g, f)-factorof G is a spanning 8ubgraph F of G satisfying g(x) < dG(z) 5 f(x) for all x E V(F). Ifg(x) = f(x) for all x E V(…     

On the Factorizations of （mg, mf）-graph

Let G be an (mg, mf)-graph, where g and f are integer-valued functions defined on V(G) and such that 0≤g(x)≤f(x) for each x ∈ V(G). It is proved that(1) If Z ≠ , both g and f may be not even, G has a (g, f)-factorization, where Z = {x ∈ V(G): mf(x)-dG(x)≤t(x) or dG(x)-mg(x)≤ t(x), t(x)= f(x)-g(x)>0}.(2) Let G be an m-regular graph with 2n vertices, m≥n. If (P1, P2,..., Pr) is a partition of m, P1 ≡ m (mod 2), Pi ≡ 0 (mod 2), i = 2,..., r, then the edge set E(G) of G can be parted into r parts E1 , E2,...,Er of E(G) such that G[Ei] is a Pi-factor of G.     

（g,f）-FACTORS WITH SPECIAL PROPERTIES IN BIPARTITE （mg,mf）-GRAPHS

Let G be a bipartite graph and g and f be two positive integer-valued functions defined on vertex set V(G) of G such that g(x)≤f(x). In this paper,some sufficient conditions related to the connectivity and edge-connectivity for a bipartite (mg,mf)-graph to have a (g,f)-factor with special properties are obtained and some previous results are generalized.Furthermore,the new results are proved to be the best possible.     

The fractional metric dimension of permutation graphs

Let G =(V(G), E(G)) be a graph with vertex set V(G) and edge set E(G). For two distinct vertices x and y of a graph G, let RG{x, y} denote the set of vertices z such that the distance from x to z is not equa l to the distance from y to z in G. For a function g defined on V(G) and for U■V(G), let g(U) =∑s∈Ug(s). A real-valued function g : V(G) → [0, 1] is a resolving function of G if g(RG{x, y}) ≥ 1 for any two distinct vertices x, y ∈ V(G). The fractional metric dimension dimf(G)of a graph G is min{g(V(G)) : g is a resolving function of G}. Let G1 and G2 be disjoint copies of a graph G, and let σ : V(G1) → V(G2) be a bijection. Then, a permutation graph Gσ =(V, E) has the vertex set V = V(G1) ∪ V(G2) and the edge set E = E(G1) ∪ E(G2) ∪ {uv | v = σ(u)}. First,we determine dimf(T) for any tree T. We show that 1 dimf(Gσ) ≤1/2(|V(G)| + |S(G)|) for any connected graph G of order at least 3, where S(G) denotes the set of support vertices of G. We also show that, for any ε 0, there exists a permutation graph Gσ such that dimf(Gσ)- 1 ε. We give examples showing that neither is there a function h1 such that dimf(G) h1(dimf(Gσ)) for all pairs(G, σ), nor is there a function h2 such that h2(dimf(G)) dimf(Gσ) for all pairs(G, σ). Furthermore,we investigate dimf(Gσ) when G is a complete k-partite graph or a cycle.     

符号混合控制问题的某些NP-完全性结果

Let G =(V, E) be a simple graph with vertex set V and edge set E. A signed mixed dominating function of G is a function f:V∪E→ {-1, 1} such that ∑_(y∈N_m(x)∪{x})f(y)≥ 1for every element x∈V∪E, where N_m(x) is the set of elements of V∪E adjacent or incident to x. The weight of f is w(f) =∑_(x∈V∪E)f(x). The signed mixed domination problem is to find a minimum-weight signed mixed dominating function of a graph. In this paper we study the computational complexity of signed mixed domination problem. We prove that the signed mixed domination problem is NP-complete for bipartite graphs, chordal graphs, even for planar bipartite graphs.     

Topological entropy of a graph map

Let G be a graph and f : G → G be a continuous map. Denote by h(f), P(f), AP(f), R(f)and ω(x, f) the topological entropy of f, the set of periodic points of f, the set of almost periodic points of f, the set of recurrent points of f and the ω-limit set of x under f, respectively. In this paper,we show that the following statements are equivalent:(1) h(f) 0.(2) There exists an x ∈ G such that ω(x, f) ∩ P(f) = ? and ω(x, f) is an infinite set.(3) There exists an x ∈ G such that ω(x, f)contains two minimal sets.(4) There exist x, y ∈ G such that ω(x, f)-ω(y, f) is an uncountable set and ω(y, f) ∩ω(x, f) = ?.(5) There exist an x ∈ G and a closed subset A ? ω(x, f) with f(A) ? A such that ω(x, f)-A is an uncountable set.(6) R(f)-AP(f) = ?.(7) f |P(f)is not pointwise equicontinuous.     

A Generalization of Orthogonal Factorizations in Graphs

Let G be a graph with vertex set V(G) and edge set E(G) and let g and f be two integer-valuated functions defined on V(G) such that g(x) ≤f(x) for all x∈V(G). Then a (g, f)-factor of G is a spanning subgraph H of G such that g(x) ≤d _H (x) ≤f(x) for all x∈V(G). A (g, f)-factorization of G is a partition of E(G) into edge-disjoint (g, f)-factors. Let = {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 ≤i≤m, has exactly r edges in common with H, then is said to be r-orthogonal to H. In this paper it is proved that every (mg + kr, mf−kr)-graph, where m, k and r are positive integers with k < m and g≥r, contains a subgraph R such that R has a (g, f)-factorization which is r-orthogonal to a given subgraph H with kr edges. This research is supported by the National Natural Science Foundation of China (19831080) and RSDP of China     

Randomly Orthogonal (g,f)-factorizations in Graphs

Let G be a graph with vertex set V(G) and edge set E(G) and let g and f be two integer-valued functions defined on V(G) such that 2k − 2 ≤ f(x) for all x ∈ V(G). Let H be a subgraph of G with mk edges. In this paper it is proved that every (mg + m − 1,mf − m + 1)-graph G has (g,f)-factorizations randomly k-orthogonal to H and shown that the result is best possible.     

Orthogonal factorizations of digraphs

Let G be a digraph with vertex set V(G) and arc set E(G) and let g = (g ⁻, g ⁺) and ƒ = (ƒ ⁻, ƒ ⁺) be pairs of positive integer-valued functions defined on V(G) such that g ⁻(x) ⩽ ƒ ⁻(x) and g ⁺(x) ⩽ ƒ ⁺(x) for each x ∈ V(G). A (g, ƒ)-factor of G is a spanning subdigraph H of G such that g ⁻(x) ⩽ id _H (x) ⩽ ƒ ⁻(x) and g ⁺(x) ⩽ od _H (x) ⩽ ƒ ⁺(x) for each x ∈ V(H); a (g, ƒ)-factorization of G is a partition of E(G) into arc-disjoint (g, ƒ)-factors. Let = {F ₁, F ₂,…, F _m} and H be a factorization and a subdigraph of G, respectively. is called k-orthogonal to H if each F _i , 1 ⩽ i ⩽ m, has exactly k arcs in common with H. In this paper it is proved that every (mg+m−1,mƒ−m+1)-digraph has a (g, f)-factorization k-orthogonal to any given subdigraph with km arcs if k ⩽ min{g ⁻(x), g ⁺(x)} for any x ∈ V(G) and that every (mg, mf)-digraph has a (g, f)-factorization orthogonal to any given directed m-star if 0 ⩽ g(x) ⩽ f(x) for any x ∈ V(G). The results in this paper are in some sense best possible.      

Existence of subgraph with orthogonal (g,f)-factorization

Simple graphs are considered. Let G be a graph andg(x) andf(x) integer-valued functions defined on V(G) withg(x)⩽f(x) for everyxɛV(G). For a subgraphH ofG and a factorizationF=|F ₁,F ₂,⃛,F ₁| ofG, if |E(H)∩E(F ₁)|=1,1⩽i⩽j, then we say thatF orthogonal toH. It is proved that for an (mg(x)+k,mf(x) -k)-graphG, there exists a subgraphR ofG such that for any subgraphH ofG with |E(H)|=k,R has a (g,f)-factorization orthogonal toH, where 1⩽k<m andg(x)⩾1 orf(x)⩾5 for everyxɛV(G). Project supported by the Chitia Postdoctoral Science Foundation and Chuang Xin Foundation of the Chinese Academy of Sciences.     

Decomposition of Graphs into (g, f)-Factors

Let G be a multigraph, g and f be integer-valued functions defined on V(G). Then a graph G is called a (g, f)-graph if g(x)≤deg _G(x)≤f(x) for each x∈V(G), and a (g, f)-factor is a spanning (g, f)-subgraph. If the edges of graph G can be decomposed into (g, f)-factors, then we say that G is (g, f)-factorable. In this paper, we obtained some sufficient conditions for a graph to be (g, f)-factorable. One of them is the following: Let m be a positive integer, l be an integer with l=m (mod 4) and 0≤l≤3. If G is an -graph, then G is (g, f)-factorable. Our results imply several previous (g, f)-factorization results. Revised: June 11, 1998     

A New Result on Alspach's Problem

Let G be a simple graph. Let g(x) and f(x) be integer-valued functions defined on V(G) with g(x)≥2 and f(x)≥5 for all x∈V(G). It is proved that if G is an (mg+m−1, mf−m+1)-graph and H is a subgraph of G with m edges, then there exists a (g,f)-factorization of G orthogonal to H. Received: January 19, 1996 Revised: November 11, 1996     

Partial Parity (g,f )-Factors and Subgraphs Covering Given Vertex Subsets

 Let G be a graph and W a subset of V(G). Let g,f:V(G)→Z be two integer-valued functions such that g(x)≤f(x) for all x∈V(G) and g(y)≡f(y) (mod 2) for all y∈W. Then a spanning subgraph F of G is called a partial parity (g,f)-factor with respect to W if g(x)≤deg _F (x)≤f(x) for all x∈V(G) and deg _F (y)≡f(y) (mod 2) for all y∈W. We obtain a criterion for a graph G to have a partial parity (g,f)-factor with respect to W. Furthermore, by making use of this criterion, we give some necessary and sufficient conditions for a graph G to have a subgraph which covers W and has a certain given property. Received: June 14, 1999?Final version received: August 21, 2000     

(g, f)-factorizations orthogonal to a subgraph in graphs

LetG be a graph with vertex setV (G) and edge setE (G), and letg andf be two integer-valued functions defined on V(G) such thatg(x)⩽(x) for every vertexx ofV(G). It was conjectured that ifG is an (mg +m - 1,mf -m+1)-graph andH a subgraph ofG withm edges, thenG has a (g,f)-factorization orthogonal toH. This conjecture is proved affirmatively. Project supported by the National Natural Science Foundation of China.     

(g,f)-Factorizations of graphs orthogonal to [1,2]-subgraph

LetG be a simple graph. Letg(x) andf(x) be integer-valued functions defined onV(G) withf(x)g(x)1 for allxV(G). It is proved that ifG is an (mg+m–1,mf–m+1)-graph andH is a [1,2]-subgraph withm edges, then there exists a (g,f)-factorization ofG orthogonal toH.This work is supported by China Postdoctoral Science Foundation and Shandong Youth Science Foundation.     

On Connected [g,f +1]-Factors in Graphs

Let G = (V (G),E(G)) be a graph with vertex set V (G) and edge set E(G), and g and f two positive integral functions from V (G) to Z⁺-{1} such that g(v) ≤ f(v) ≤ d_G(v) for all v ∈V (G), where d_G(v) is the degree of the vertex v. It is shown that every graph G, including both a [g,f]-factor and a hamiltonian path, contains a connected [g,f +1]-factor. This result also extends Kano’s conjecture concerning the existence of connected [k,k+1]-factors in graphs. * The work of this author was supported by NSFC of China under Grant No. 10271065, No. 60373025. † The work of these authors was also supported in part by the US Department of Energy’s Genomes to Life program (http://doegenomestolife.org/) under project, “Carbon Sequestration in Synechococcus sp.: From Molecular Machines to Hierarchical Modeling” (www.genomes2life.org) and by National Science Foundation (NSF/DBI-0354771,NSF/ITR-IIS-0407204).