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.

Cycle Multiplicity of Total Graph of Complete Bipartite Graph

Cycle multiplicity of a graph G is the maximum number of edge disjoint cycles in G. In this paper, we determine the cycle multiplicity of and then obtain the formula of cycle multiplicity of total graph of complete bipartite graph, this generalizes the result for, which is given by M.M. Akbar Ali in [1].

Vertex-distinguishing E-total Coloring of Complete Bipartite Graph K 7,n when7≤n≤95

Let G be a simple graph. A total coloring f of G is called an E-total coloring if no two adjacent vertices of G receive the same color, and no edge of G receives the same color as one of its endpoints. For an E-total coloring f of a graph G and any vertex x of G, let C（x） denote the set of colors of vertex x and of the edges incident with x, we call C（x） the color set of x. If C（u） ≠ C（v） for any two different vertices u and v of V （G）, then we say that f is a vertex-distinguishing E-total coloring of G or a VDET coloring of G for short. The minimum number of colors required for a VDET coloring of G is denoted by Хvt^e（G） and is called the VDE T chromatic number of G. The VDET coloring of complete bipartite graph K7,n （7 ≤ n ≤ 95） is discussed in this paper and the VDET chromatic number of K7,n （7 ≤ n ≤ 95） has been obtained.

Two conditions for a bipartite graph to be a k-deleted graph

A k-regular spanning subgraph of graph G is called a k-factor of G. Graph G is called a k-deleted graph if G-e has a k-factor for each edge e. A graph G=(X,Y) with bipartition (X,Y) is called a bipartite graph if every edge of G has one endpoint in X and the other in Y.It is proved that a bipartite graph G=(X,Y) with X=Y is a k-deleted graph if and only if kS≤r 1+2r 2+...+k(r k+...+r Δ)-ε(S) for all SX. Using this result we give a sufficient neighborhood condition for a bipartite to be a k-deleted graph.

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.

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.

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)) .

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.

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.

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.

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.

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.

On Enomoto's problems in a bipartite graph

In this paper, we obtain the following result: Let k, n 1 and n 2 be three positive integers, and let G = (V 1,V 2;E) be a bipartite graph with |V1| = n 1 and |V 2| = n 2 such that n 1 ? 2k + 1, n 2 ? 2k + 1 and |n 1 ? n 2| ? 1. If d(x) + d(y) ? 2k + 2 for every x ∈ V 1 and y ∈ V 2 with xy $ \notin $ E(G), then G contains k independent cycles. This result is a response to Enomoto’s problems on independent cycles in a bipartite graph.

On 2-Factors with Prescribed Properties in a Bipartite Graph

Liu and Yan gave the degree condition for a balanced bipartite graph G = （V1, V2; E） to have k vertex-disjoint quadrilaterals containing any given k independent edges e1,……, ek of G, respectively. They also conjectured that for any k independent edges e1,……, ek of G, G has a 2-factor with k cycles C1, C2, ……, Ck with respect to {e1, e2,……, ek} such that k - 1 of them are quadrilaterals. In this paper, we prove this conjecture.

Bipartite graph-based control flow checking for COTS-based small satellites

Abstract Single event upset （SEU） effect, caused by highly energized particles in aerospace, threatens the reliability and security of small satellites composed of commercialofftheshelves （COTS）. SEU induced control flow errors （CFEs） may cause unpredictable behavior or crashes of COTSbased small satellites. This paper proposes a generic softwarebased control flow checking technique （CFC） and bipartite graphbased control flow checking （BGCFC）. To simplify the types of illegal branches, it transforms the conventional control flow graph into the equivalent bipartite graph. It checks the legal ity of control flow at runtime by comparing a global signature with the expected value and introduces consecutive IDs and bitmaps to reduce the time and memory overhead. Theoretical analysis shows that BGCFC can detect all types of internode CFEs with constant time and memory overhead. Practical tests verify the result of theoretical analysis. Compared with previous techniques, BGCFC achieves the highest error detection rate, lower time and memory overhead; the composite result in evaluation fac tor shows that BGCFC is the most effective one among all these techniques. The results in both theory and practice verify the applicability of BGCFC for COTSbased small satellites.

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.

Graham's pebbling conjecture on product of complete bipartite graphs

The pebbling number of a graph G,f(G),is the least n such that,no matter how n pebbles are placed on the vertices of G,we can move a pebble to any vertex by a sequence of moves,each move taking two pebbles off one vertex and placing one on an adjacent vertex.Graham conjectured that for any connected graphs G and H,f(G×H)≤f(G)f(H).We show that Graham's conjecture holds true of a complete bipartite graph by a graph with the two-pebbling property.As a corollary,Graham's conjecture holds when G and H are complete bipartite graphs.