Orthogonal one-factorizations of complete multipartite graphs

A k-colouring of a graph G with colours 1 , 2 , … , k is canonical with respect to an ordering π = v 1 , v 2 , … , v n of the vertices of G if adjacent vertices are assigned different colours and, for 1 ≤ c ≤ k , whenever colour c is assigned to a vertex v i , each colour less than c has been assigned to a vertex that precedes v i in π . The canonical k-colouring graph of G with respect to π is the graph Can k π ( G ) with vertex set equal to the set of canonical k-colourings of G with respect to π , with two of these being adjacent if and only if they differ in the colour assigned to exactly one vertex. Connectivity and Hamiltonicity of canonical colouring graphs of bipartite and complete multipartite graphs is studied. It is shown that for complete multipartite graphs, and bipartite graphs there exists a vertex ordering π such that Can k π ( G ) is connected for large enough values of k. It is proved that a canonical colouring graph of a complete multipartite graph usually does not have a Hamilton cycle, and that there exists a vertex ordering π such that Can k π ( K m , n ) has a Hamilton path for all k ≥ 3 . The paper concludes with a detailed consideration of Can k π ( K 2 , 2 , … , 2 ) . For each k ≥ χ and all vertex orderings π , it is proved that Can k π ( K 2 , 2 , … , 2 ) is either disconnected or isomorphic to a particular tree.

In this paper we present some results on the sequence of coefficients of the chromatic polynomial of a graph relative to the complete graph basis, that is, when it is expressed as the sum of the chromatic polynomials of complete graphs. These coefficients are the coefficients of what is often called the σ-polynomial. We obtain necessary and sufficient conditions for this sequence to be symmetrical, and we prove that it is ‘skewed’ and decreasing beyond its midpoint. We also prove that it is strongly log-concave when G is a complete multipartite graph. © 1996 John Wiley & Sons, Inc.

The Shapley distance in a graph is defined based on Shapley value in cooperative game theory. It is used to measure the cost for a vertex in a graph to access another vertex. In this paper, we establish the Shapley distance between two arbitrary vertices for some special graphs, i.e., path, tree, cycle, complete graph, complete bipartite, and complete multipartite graph. Moreover, based on the Shapley distance, we propose a new index, namely Shapley index, and then compare Shapley index with Wiener index and Kirchhoff index for these special graphs. We also characterize the extremal graphs in which these three indices are equal.

Lin et al. (2013) [16] conjectured that, the complete multipartite graphs are determined by their distance spectra. This was confirmed by Jin and Zhang (2014) [14]. In the present paper we give a shorter and more readable proof of the conjecture.

Let the star on n vertices, namely K 1, n − 1 be denoted by S n . If every two coloring of the edges of a complete balanced multipartite graph K j × s there is a copy of S n in the first color or a copy of S m in the second color, then we will say K j × s → ( S n , S m ) . The size Ramsey multipartite number m j ( S n , S m ) is the smallest natural number s such that K j × s → ( S n , S m ) . In this paper, we obtain the exact values of the size Ramsey numbers m j ( S n , S m ) for n , m ≥ 3 and j ≥ 3 .

The Tur\'{a}n number for a graph $H$ is the least possible number of edges on an $n$-vertex graph with no copy of $H$ as a subgraph. For graphs $G$ and $H$, the \emph{anti-Ramsey number}, denoted $\ar(G,H)$, is the minimum number of colors $d$ such that for any edge coloring with $d$ colors there exists a rainbow copy of $H$. The concept of anti-Ramsey theory was first introduced by Erd\H{o}s, Simonovits, and S\'{o}s in 1973 as a dual problem to traditional Ramsey theory. This thesis investigates Tur\'{a}n theory and anti-Ramsey theory for specific graph families, including cliques in multipartite graphs and matchings in hypergraphs. In Chapter 2, the anti-Ramsey number for disjoint cliques in complete, balanced multipartite graphs is given for certain sizes of cliques and multipartite graphs. In particular, the value $\ar(K_{r\times n}, nK_r)$ is given for all integers $n,r\ge 3$ and $\ar(K_{n,n,n}, 3K_r)$ is given for all integers $n,r\ge 3$. Chapter 3 generalizes Tur\'{a}n and anti-Ramsey theory to the context of hypergraphs. The Tur\'{a}n number for $3$-matchings in complete $3$-uniform hypergraphs is determined. Next, two proofs of the value for the anti-Ramsey number for $4$-matchings in complete hypergraphs are presented. One relies on the Tur\'{a}n number for $3$-matchings, and the other utilizes a method called progressive induction. Chapter 4 features additional work related to combinatorics more generally.

Complete Multipartite Graphs and their Null Set

Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that the continuous-time quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs that have the instantaneous exactly uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs of the symmetric group lack this mixing property as well.

The aggregate of the absolute values of the graph eigenvalues is called the energy of a graph. It is used to approximate the total _-electron energy of molecules. Thus, finding a new mechanism to calculate the total energy of some graphs is a challenge; it has received a lot of research attention. We study the eigenvalues of a complete tripartite graph Ti,i,n−2i , for n _ 4, based on the adjacency, Laplacian, and signless Laplacian matrices. In terms of the degree sequence, the extreme eigenvalues of the irregular graphs energy are found to characterize the component with the maximum energy. The chemical HMO approach is particularly successful in the case of the total _-electron energy. We showed that some chemical components are equienergetic with the tripartite graph. This discovering helps easily to derive the HMO for most of these components despite their different structures.

A graph Γ is called a Deza graph if it is regular and the number of common neighbors of any two distinct vertices is one of two fixed values. A Deza graph is called a strictly Deza graph if it has diameter 2 and is not strongly regular. In 1992, Gardiner et al. proved that a strongly regular graph that contains a vertex with disconnected second neighborhood is a complete multipartite graph with parts of the same size greater than 2. In this paper, we study strictly Deza graphs with disconnected second neighborhoods of vertices. In Section 2, we prove that, if each vertex of a strictly Deza graph has disconnected second neighborhood, then the graph is either edge-regular or coedge-regular. In Sections 3 and 4, we consider strictly Deza graphs that contain at least one vertex with disconnected second neighborhood. In Section 3, we show that, if such a graph is edge-regular, then it is the s-coclique extension of a strongly regular graph with parameters (n, k, λ, μ), where s is an integer, s ≥ 2, and λ = μ. In Section 4, we show that, if such a graph is coedge-regular, then it is the 2-clique extension of a complete multipartite graph with parts of the same size greater than or equal to 3.

Let G be a graph and let s be a vertex of G. We consider the structure of the set of all lifts of two edges incident with s that preserve edge-connectivity. Mader proved that two mild hypotheses imply there is at least one pair that lifts, while Frank showed (with the same hypotheses) that there are at least $$(\deg (s)-1)/2$$(deg(s)-1)/2 disjoint pairs that lift. We consider the lifting graph: its vertices are the edges incident with s, two being adjacent if they form a liftable pair. We have three main results, the first two with the same hypotheses as for Mader's Theorem. (i) Let F be a subset of the edges incident with s. We show that F is independent in the lifting graph of G if and only if there is a single edge-cut C in G of size at most $$r+1$$r+1 containing all the edges in F, where r is the maximum number of edge-disjoint paths from a vertex (not s) in one component of $$G-C$$G-C to a vertex (not s) in another component of $$G-C$$G-C. (ii) In the k-lifting graph, two edges incident with s are adjacent if their lifting leaves the resulting graph with the property that any two vertices different from s are joined by k pairwise edge-disjoint paths. If both $$\deg (s)$$deg(s) and k are even, then the k-lifting graph is a connected complete multipartite graph. In all other cases, there are at most two components. If there are exactly two components, then each component is a complete multipartite graph. If $$\deg (s)$$deg(s) is odd and there are two components, then one component is a single vertex. (iii) Huck proved that if k is odd and G is $$(k+1)$$(k+1)-edge-connected, then G is weakly k-linked (that is, for any k pairs $$\{x_i,y_i\}$${xi,yi}, there are k edge-disjoint paths $$P_i$$Pi, with $$P_i$$Pi joining $$x_i$$xi and $$y_i$$yi). We use our results to extend a slight weakening of Huck's theorem to some infinite graphs: if k is odd, every $$(k+2)$$(k+2)-edge-connected, locally finite, 1-ended, infinite graph is weakly k-linked.

In this paper we completely solve the problem of finding a maximum packing of any complete multipartite graph with edge-disjoint 4-cycles, and the minimum leaves are explicitly given. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 107–127, 2001

Labelling problems are very useful in modelling, solving problems in other areas and enhance the study of algebraic structures. Nowadays, a more practical, accurate and efficient way of labeling a graph is using some mathematical methods like integer programming or constraint programming. This will improve the scope and quality of research. Here, considering the applications of Graceful and Harmonious labelings, we introduce a few new types of labeling, namely, gracefully harmonious labeling, gracefully felicitous labeling, gracefully elegant labeling, gracefully strongly c-elegant labeling and gracefully semi-harmonious labeling. In this paper, we prove that some complete multi-partite graphs is gracefully harmonious, gracefully felicitous and gracefully semi-harmonious. Keywords: Labeling; multi-partite graph; gracefully harmonious; gracefully felicitous; gracefully semi-harmonious. 2010 Mathematics Subject Classification: 05C78

Maximizing or minimizing the number of copies of a fixed graph in a large host graph is one of the most classical topics in extremal graph theory. Indeed, one of the most famous problems in extremal graph theory, the Erdős-Rademacher problem, which can be traced back to the 1940s, asks to determine the minimum number of triangles in a graph with a given number of vertices and edges. It was conjectured that the mnimum is attained by complete multipartite graphs with all parts but one of the same size whilst the remaining part may be smaller. The problem was widely open in the regime of four or more parts until Razborov resolved the problem asymptotically in 2008 as one of the first applications of his newly developed flag algebra method. This catalyzed a line of research on the structure of extremal graphs and extensions. In particular, Reiher asymptotically solved in 2016 the conjecture of Lovász and Simonovits from the 1970s that the same graphs are also minimizers for cliques of arbitrary size. This paper deals with a problem concerning the opposite direction: _What is the maximum number of triangles in a graph with a given number $n$ of vertices and a given maximum degree $D$?_ Gan, Loh and Sudakov in 2015 conjectured that the graph maximizing the number of triangles is always a union of disjoint cliques of size $D+1$ and another clique that may be smaller, and showed that if such a graph maximizes the number of triangles, it also maximizes the number of cliques of any size $r\ge 4$. The author presents a remarkably simple and elegant argument that proves the conjecture exactly for all $n$ and $D$.

A graph is a multiclique if its connected components are cliques. A graph is a complete multipartite graph if it is the complement of a multiclique. A graph is a multiclique-multipartite graph if its vertex set has a partition U, W such that G(U) is complete multipartite, G(W) is a multiclique and every two vertices u(U, v(W are adjacent. We describe a polynomial time algorithm to find in polygon-circle graphs a maximum induced complete multipartite subgraph containing an induced K 2,2. In addition, we describe polynomial time algorithms to find maximum induced multicliques and multiclique-multipartite subgraphs in circle graphs. These problems have applications for clustering of proteins by PPI criteria.Keywordspolygon-circle graphcircle graphinduced multicliqueinduced complete multipartite subgraphProtein-Protein-Interaction