Complete Partite Graphs Research Articles

Stability from graph symmetrisation arguments with applications to inducibility

AbstractWe present a sufficient condition for the stability property of extremal graph problems that can be solved via Zykov's symmetrisation. Our criterion is stated in terms of an analytic limit version of the problem. We show that, for example, it applies to the inducibility problem for an arbitrary complete bipartite graph , which asks for the maximum number of induced copies of in an ‐vertex graph, and to the inducibility problem for and , the only complete partite graphs on at most five vertices for which the problem was previously open.

Equitable Coloring On Rooted Product Of Graphs

A finite and simple graph is said to be equitably colorable if its vertices can be partitioned into classes such that each is an independent set and holds for every . The smallest integer for which is equitable chromatic number of and denoted by . The equitable chromatic threshold of a graph , denoted by , is the minimum such that is equitably colorable for all . This paper focuses on the equitable colorability of rooted product of graphs, in particular, exact values or upper bounds of and when and are cycles, paths, complete graphs and complete partite graphs have been found.

Bivariate chromatic polynomials in computer algebra

The computation of the chromatic polynomial of a graph, which was introduced by Birkhoff (1912), is an NP-complete problem. Consequently, this is also valid for the bivariate generalization of the chromatic polynomial by Dohmen et al. (2003). A recursion formula, which was presented by Averbouch et al. (2008), has exponential complexity. Hence, the aim of the current article is to find efficient algorithms or formulas for the calculation of the bivariate chromatic polynomial for special types of graphs. The following results will be presented:We found efficient formulas for complete graphs, from which the edges of stars with pairwise different vertices are deleted, for complete partite graphs and for special split graphs.Furthermore, in a section about separators, we show that the special case of separating complete subgraphs, which is very simple in the univariate case, requires rather complex methods in the bivariate case.Finally, we establish a connection between the bivariate chromatic polynomial and the matching polynomial for complete graphs from which the edges of stars with pairwise different vertices are removed, as well as for bi-cliques.Some algorithms are implemented in Mathematica, and examples are presented.

Weighted antimagic labeling

A graph G=(V,E) is weighted-k-antimagic if for each w:V→R, there is an injective function f:E→{1,…,|E|+k} such that the following sums are all distinct: for each vertex u, ∑v:uv∈Ef(uv)+w(u). When such a function f exists, it is called a (w,k)-antimagic labeling of G. A connected graph G is antimagic if it has a (w0,0)-antimagic labeling, for w0(u)=0, for each u∈V.In this work, we prove that all the complete bipartite graphs Kp,q, are weighted-0-antimagic when 2≤p≤q and q≥3. Moreover, an algorithm is proposed that computes in polynomial time a (w,0)-antimagic labeling of the graph. Our result implies that if H is a complete partite graph, with H≠K1,q, K2,2, then any connected graph G containing H as a spanning subgraph is antimagic.

Weighted antimagic labeling: an algorithmic approach

A graph G=(V,E) is weighted-k-antimagic if for each w:V→R, there is an injective function f:E→{1,…,|E|+k} such that for each vertex u the following sums are all distinct: ∑v:uv∈Ef(uv)+w(u). When such a function f exists, it is called a (w,k)-antimagic labeling of G. A connected graph G is antimagic if it has a (w,0)-antimagic labeling where w(u)=0 for each u∈V.In this work, an algorithm is proposed that computes in polynomial time a (w,0)-antimagic labeling of the complete bipartite graph Km,n=(V,E), when 2≤n≤m and m≥3, for any function w:V→R.On the one hand, the existence of this labeling provides new support to a well-known conjecture of Hartsfield and Ringel asserting that any connected graph with at least two edges is antimagic. More precisely, for each complete partite graph H, with H∉{K1,m,K2,2} we prove that any connected graph G containing H as an spanning subgraph is weighted-0-antimagic. Moreover, given a function w, a (w,0)-antimagic labeling can be computed in polynomial time. On the other hand, the ideas used in the design of our algorithm allows us to give a short algorithmic proof of the following fact: each connected graph G on n vertices having a universal vertex is weighted-1-antimagic, unless G=K1,n−1 and n is even.

Counting Links and Knots in Complete Graphs

We investigate the minimal number of links and knots in complete partite graphs. We provide exact values or bounds on the minimal number of links for all complete partite graphs with all but 4 vertices in one partition, or with 9 vertices in total. In particular, we find that the minimal number of links for $K_{4,4,1}$ is 74. We also provide exact values or bounds on the minimal number of knots for all complete partite graphs with 8 vertices.

An approximate blow-up lemma for sparse pseudorandom graphs

We state a sparse approximate version of the blow-up lemma, showing that regular partitions in sufficiently pseudorandom graphs behave almost like complete partite graphs for embedding graphs with maximum degree Δ. We show that (p,γ)-jumbled graphs, with γ=o(pmax(2Δ,Δ+3/2)n), are “sufficiently pseudorandom”.The approach extends to random graphs Gn,p with p≫(lognn)1/Δ.

An algorithm for finding a maximum [formula omitted]-matching excluding complete partite subgraphs

For an integer t and a fixed graph H, we consider the problem of finding a maximum t-matching not containing H as a subgraph, which we call the H-free t-matching problem. This problem is a generalization of the problem of finding a maximum 2-matching with no short cycles, which has been well-studied as a natural relaxation of the Hamiltonian circuit problem. When H is a complete graph Kt+1 or a complete bipartite graph Kt,t, in 2010, Bérczi and Végh gave a polynomial-time algorithm for the H-free t-matching problem in simple graphs with maximum degree at most t+1. A main contribution of this paper is to extend this result to the case when H is a t-regular complete partite graph. We also show that the problem is NP-complete when H is a connected t-regular graph that is not complete partite. Since it is known that, for a connected t-regular graph H, the degree sequences of all H-free t-matchings in a graph form a jump system if and only if H is a complete partite graph, our results show that the polynomial-time solvability of the H-free t-matching problem is consistent with this condition.

The saturation function of complete partite graphs

A graph G is called F-saturated if it is F-free but the addition of any missing edge to G creates a copy of F. Let the saturation function sat(n, F) be the minimum number of edges that an F-saturated graph on n vertices can have. We determine this function asymptotically for every fixed complete partite graph F as n tends to infinity and give some structural information about almost extremal F-saturated graphs. If the two largest parts of F have different sizes, then we can reduce the error term to an additive constant.

Dense graphs are antimagic

AbstractAn antimagic labeling of graph a with m edges and n vertices is a bijection from the set of edges to the integers 1,…,m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. A conjecture of Ringel (see 4) states that every connected graph, but K2, is antimagic. Our main result validates this conjecture for graphs having minimum degree Ω (log n). The proof combines probabilistic arguments with simple tools from analytic number theory and combinatorial techniques. We also prove that complete partite graphs (but K2) and graphs with maximum degree at least n – 2 are antimagic. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 297–309, 2004

Steiner trade spectra of complete partite graphs

The Steiner trade spectrum of a simple graph G is the set of all integers t for which there is a simple graph H whose edges can be partitioned into t copies of G in two entirely different ways. The Steiner trade spectra of complete partite graphs were determined in all but a few cases in a recent paper by Billington and Hoffman (Discrete Math. 250 (2002) 23). In this paper we resolve the remaining cases.

Trade spectra of complete partite graphs

The trade spectrum of a graph G is essentially the set of all integers t for which there is a graph H whose edges can be partitioned into t copies of G in two entirely different ways. In this paper we determine the trade spectrum of complete partite graphs, in all but a few cases.

Edge domination in complete partite graphs

An edge dominating set in a graph G is a set of edges D such that every edge not in D is adjacent to an edge of D. An edge domatic partition of a graph G=( V, E) is a collection of pairwise-disjoint edge dominating sets of G whose union is E. The maximum size of an edge domatic partition of G is called the edge domatic number. In this paper, we study the edge domatic number of the complete partite graphs and give the answers for balanced complete partite graphs and complete split graphs.

Triangular embeddings of K((i‐2) n, n,…,n)

AbstractFor complete i‐partite graphs of the form K(n1, n, n, …, n) the largest value of n1 that allows the graph to be triangularly‐embedded into a surface is (i‐2)n. In this paper the author constructs triangular embeddings into surfaces of some complete partite graphs of the form K((i‐2)n, n, …, n). The embeddings are exhibited using embedding schemes but the surfaces into which K((i‐2)n, n, …, n) are triangularly embedded can be seen to be particularly nice branched covers of a surface into which K(i‐2, 1, 1,…,1) is triangularly embedded.

A note on circular dimension

In [1] Feinberg conjectures that the maximum circular dimension of all graphs having n vertices is attained by a complete partite graph. In this note we show that this is not so.

The circular dimension of a graph

A graph is a pair ( V, I), V being the vertices and I being the relation of adjacency on V. Given a graph G, then a collection of functions { f i } m n= 1 , each f i mapping each vertex of V into anarc on a fixed circle, is said to define an m-arc intersection model for G if for all x,y ϵ V, xly ⇔ (∨ i ⩽ m)( f i ( x)∩ f i ( y)≠Ø). The circular dimension of a graph G is defined as the smallest integer m such that G has an m-arc intersection model. In this paper we establish that the maximum circular dimension of any complete partite graph having n vertices is the largest integer p such that 2 p + p⩽ n+1.