Perfect 1-k Matchings of Bipartite Graphs

Let k be a positive integer and G a bipartite graph with bipartition (X,Y). A perfect 1-k matching is an edge subset M of G such that each vertex in Y is incident with exactly one edge in M and each vertex in X is incident with exactly k edges in M. A perfect 1-k matching is an optimal semi-matching related to the load-balancing problem, where a semi-matching is an edge subset M such that each vertex in Y is incident with exactly one edge in M, and a vertex in X can be incident with an arbitrary number of edges in M. In this paper, we give three sufficient and necessary conditions for the existence of perfect 1-k matchings and for the existence of 1-k matchings covering | X |−dvertices in X, respectively, and characterize k-elementary bipartite graph which is a graph such that the subgraph induced by all k-allowed edges is connected, where an edge is k-allowed if it is contained in a perfect 1-k matching.

Vertex-distinguishing IE-total Colorings of Complete Bipartite Graphs K8,n

Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. For each vertex x of G, let C(x) be the set of colors of vertex x and edges incident to x under f. For an IE-total coloring f of G using k colors, if C(u) ≠ C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-coloring of G or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χ_(vt)^(ie) (G) and is called vertex-distinguishing IE-total chromatic number or the VDIET chromatic number of G for short. The VDIET colorings of complete bipartite graphs K_(8,n)are discussed in this paper. Particularly, the VDIET chromatic number of K_(8,n) are obtained.

Signed Roman (Total) Domination Numbers of Complete Bipartite Graphs and Wheels

A signed(res. signed total) Roman dominating function, SRDF(res.STRDF) for short, of a graph G =(V, E) is a function f : V → {-1, 1, 2} satisfying the conditions that(i)∑v∈N[v]f(v) ≥ 1(res.∑v∈N(v)f(v) ≥ 1) for any v ∈ V, where N [v] is the closed neighborhood and N(v) is the neighborhood of v, and(ii) every vertex v for which f(v) =-1 is adjacent to a vertex u for which f(u) = 2. The weight of a SRDF(res. STRDF) is the sum of its function values over all vertices.The signed(res. signed total) Roman domination number of G is the minimum weight among all signed(res. signed total) Roman dominating functions of G. In this paper,we compute the exact values of the signed(res. signed total) Roman domination numbers of complete bipartite graphs and wheels.

ORTHOGONAL (g,f)-FACTORIZAFIONS OF 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 positive integer-valued functions defined on V(G) such that g(x) ≤ f(x) for every vertex x of V(G). Then a (g, f)-factor of G is a spanning subgraph H of G such that g(x) ≤ dH(x) 5 f(x) for each x ∈ V(H). A (g, f)-factorization of G is a partition of E(G) into edge-disjoint (g, f)-factors. Let F = {F1, F2,…… ， Fm } and H be a factorization and a subgraph of G, respectively. If F, 1 ≤ i ≤ m， has exactly one edge in common with H, then it is said that ■ is orthogonal to H. It is proved that every bipartite (mg + m - 1, mf - m + 1 )-graph G has a (g, f)-factorization orthogonal to k vertex disjoint m-subgraphs of G if 2-k ≤ g(x) for all x ∈ V(G). Furthermore, it is showed that the results in this paper are best possible.

K_(1,p)~k-FACTORIZATION OF COMPLETE BIPARTITE GRAPHS

In this paper, it is shown that a sufficient condition for the existence of a K 1,p k factorization of K m,n , whenever p is a prime number and k is a positive integer, is (1) m≤p kn,(2) n≤p km,(3)p kn-m≡p km-n ≡0(mod( p 2k -1 )) and (4) (p kn-m)(p km-n) ≡0(mod( p k -1)p k×(p 2k -1)(m+n)) .

Bipartite Graphs with the First and Second Largest Laplacian Spectral Radius

Let Bn^k be the class of bipartite graphs with n vertices and k cut edges. The extremal graphs with the first and the second largest Laplacian spectral radius among all graphs in Bn^K are presented. The bounds of the Laplacian spectral radius of these extremal graphs are also obtained.

The Chromatic Uniqueness of Bipartite Graphs K(m,n)-A with |A|=2

The chromatically uniqueness of bipartite graphs K （m, n） - A（]A] = 2） was studied. With comparing the numbers of partitions into r color classes of two chromatically equivalent graphs, one general numerical condition guaranteeing that K（ m, n） - A （ I A ] = 2） is chromatically unique were obtained. This covers and improves the former correlative results.

The Further Results of the Chromatic Uniqueness of Certain Bipartite Graphs K(m, n)-A

With its comprehensive application in network information engineering （e. g. dynamic spectrum allocation under different distance constraints ） and in network combination optimization （e. g. safe storage of deleterious materials）, the graphs＇ cloring theory and chromatic uniqueness theory have been the forward position of graph theory research. The later concerns the equivalent classification of graphs with their color polynomials and the determination of uniqueness of some equivalent classification under isomorphism. In this paper, by introducing the concept of chromatic normality and comparing the number of partitions of two chromatically equivalent graphs, a general numerical condition guarenteeing that bipartite graphs K （ m, n） - A （A belong to E（K （m, n） ） and ｜ A ｜≥ 2） is chromatically unique was obtained and a lot of chromatic uniqueness graphs of bipartite graphs K（m, n） - A were determined. The results obtained in this paper were general. And the results cover and extend the majority of the relevant results obtained within the world.

On the Wiener Index of the Complements of Bipartite Graphs

The Wiener index W（G） of a graph G is defined as the sum of distances between all pairs of vertices of the graph, Let G＊c, is the set of the complements of bipartite graphs with order n. In this paper, we characterize the graphs with the maximum and second-maximum Wiener indices among all the graphs in G＊c, respectively.

E-Total Coloring of Complete Bipartite Graphs K_(5,n)(5≤n≤7 113)Which Are Vertex-Distinguished by Multiple Sets

In this study,using the method of contradiction and the pre-assignment of chromatic sets,we discuss the E-total coloring of complete bipartite graphs K_(5,n)(5≤n≤7 113) which are vertex-distinguished by multiple sets.The vertex-distinguishing E-total chromatic numbers of this kind of graph are determined.

On the Turán Numbers of Linear Forests in Bipartite Graphs

A linear forest is a graph consisting of paths.In this paper,the authors determine the maximum number of edges in an(m,n)-bipartite graph which does not contain a linear forest consisting of paths on at least four vertices for n≥m when m is sufficiently large.

Core Decomposition and Maintenance in Bipartite Graphs

The prevalence of graph data has brought a lot of attention to cohesive and dense subgraph mining.In contrast with the large number of indexes proposed to help mine dense subgraphs in general graphs,only very few indexes are proposed for the same in bipartite graphs.In this work,we present the index called˛.ˇ/-core number on vertices,which reflects the maximal cohesive and dense subgraph a vertex can be in,to help enumerate the(α,β)-cores,a commonly used dense structure in bipartite graphs.To address the problem of extremely high time and space cost for enumerating the(α,β)-cores,we first present a linear time and space algorithm for computing the˛.ˇ/-core numbers of vertices.We further propose core maintenance algorithms,to update the core numbers of vertices when a graph changes by avoiding recalculations.Experimental results on different real-world and synthetic datasets demonstrate the effectiveness and efficiency of our algorithms.

Weak External Bisection of Some Graphs

Let G be a graph. A bipartition of G is a bipartition of V (G) with V (G) = V1 ∪ V2 and V1 ∩ V2 = ∅. If a bipartition satisfies ∥V1∣ - ∣V2∥ ≤ 1, we call it a bisection. The research in this paper is mainly based on a conjecture proposed by Bollobás and Scott. The conjecture is that every graph G has a bisection (V1, V2) such that ∀v ∈ V1, at least half minuses one of the neighbors of v are in the V2;∀v ∈ V2, at least half minuses one of the neighbors of v are in the V1. In this paper, we confirm this conjecture for some bipartite graphs, crown graphs and windmill graphs.

VERTEX-DISJOINT QUADRILATERALS IN BIPARTITE GRAPHS

H. Wang considered the minimum degrees condition that G has largevertex-disjoint cycles in bipartite graphs. Motivated by this, we consider the small vertex-disjointcycles in bipartite graphs in this paper. We prove the following result: Let m ≥ 3, n ≥ 2 and k≥ 1 be three integers. Let G = (V_1,V_2;E) be a bipartite graph with |V_1| = |V_2| = n ≥ 2k+1.

Topological Minors in Bipartite Graphs

For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and for integers s andt such that 2≤ s ≤ t, 0≤ m-s ≤ n-t, andre＋n≤ 2s＋t-1, we prove that if G has at least mn- （2（m - s） ＋ n - t） edges then it contains a subdivision of the complete bipartite K（s,t） with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn- （2（m - s） ＋ n - t ＋ 1） edges for this topological Turan type problem.

Bipartite Graphs with the Maximum Sum of Squares of Degrees

In this paper we determine all the bipartite graphs with the maximum sum of squares of degrees among the ones with a given number of vertices and edges.

Bipartite Graphs K_(n,n+r)-A(|A|≤3) Determined by Their Cycle Length Distributions

The cycle length distribution of a graph G of order n is a sequence （c1 （G）,…, cn （G））, where ci （G） is the number of cycles of length i in G. In general, the graphs with cycle length distribution （c1（G） ,…,cn（G）） are not unique. A graph G is determined by its cycle length distribution if the graph with cycle length distribution （c1 （G）,…, cn （G）） is unique. Let Kn,n＋r be a complete bipartite graph and A lohtaib in E（Kn,n＋r）. In this paper, we obtain： Let s 〉 1 be an integer. （1） If r = 2s, n 〉 s（s - 1） ＋ 2｜A｜, then Kn,n＋r - A （A lohtain in E（Kn,n＋r）,｜A｜ ≤ 3） is determined by its cycle length distribution; （2） If r = 2s ＋ 1,n 〉 s^2 ＋ 2｜A｜, Kn,n＋r - A （A lohtain in E（Kn,n＋r）, ｜A｜ ≤3） is determined by its cycle length distribution.

Solving the Maximum Matching Problem on Bipartite Star123-Free Graphs in Linear Time

The bipartite Star123-free graphs were introduced by V. Lozin in [1] to generalize some already known classes of bipartite graphs. In this paper, we extend to bipartite Star123-free graphs a linear time algorithm of J. L. Fouquet, V. Giakoumakis and J. M. Vanherpe for finding a maximum matching in bipartite Star123, P7-free graphs presented in [2]. Our algorithm is a solution of Lozin’s conjecture.

K_(p,q)-factorization of complete bipartite graphs

Let Km,n be a completebipartite graph with two partite sets having m and n vertices,respectively. A Kp,q-factorization of Km,n is a set ofedge-disjoint Kp,q-factors of Km,n which partition theset of edges of Km,n. When p=1 and q is a prime number,Wang, in his paper 'On K1,k-factorizations of a completebipartite graph' (Discrete Math, 1994, 126: 359-364),investigated the K1,q-factorization of Km,n and gave asufficient condition for such a factorization to exist. In the paper'K1,k-factorizations of complete bipartite graphs' (DiscreteMath, 2002, 259: 301-306), Du and Wang extended Wang's resultto the case that q is any positive integer. In this paper, we give a sufficient condition for Km,n to have aKp,q-factorization. As a special case, it is shown that theMartin's BAC conjecture is true when p:q=k:(k+1) for any positiveinteger k.

A Characterization of PM-compact Hamiltonian Bipartite Graphs

The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings in G. A graph is called perfect matching compact（shortly, PM-compact）, if its perfect matching polytope has diameter one. This paper gives a complete characterization of simple PM-compact Hamiltonian bipartite graphs. We first define two families of graphs, called the H2C-bipartite graphs and the H23-bipartite graphs, respectively. Then we show that, for a simple Hamiltonian bipartite graph G with ｜V（G）｜ ≥ 6, G is PM-compact if and only if G is K3,3, or G is a spanning Hamiltonian subgraph of either an H2C-bipartite graph or an H23-bipartite graph.