Decompositions of Complete Multipartite Graphs into Complete Graphs (original) (raw)

On the cyclic decomposition of complete graphs into bipartite graphs

2001

Let G be a graph with n edges. It is known that there exists a cyelic Gdecomposition of K 2n+1 if and only if G has a p-Iabeling. An a-labeling of G easily yields both a cyelic G-decomposition of Kn,n and of K2nx+l for all positive integers x. It is well-known that certain classes of bipartite graphs (including certain trees) do not have a-Iabelings. :Moreover, there are bipartite graphs with n edges which do not cyclically divide Kn,n. In this article, \ve introduce the concept of an ordered p-labeling (denoted by p+) of a bipartite graph, and prove that if a graph G with n edges has a p+ -labeling, then there is a cyclic G-decomposition of K 2nx+1 for all positive integers .17. vVe also introduce the concept of a O-labeling which is a more restrictive p+ -labeling. We conjecture that all forests have a p+labeling and show that the vertex-disjoint union of any finite collection of graphs that admit a-Iabelings admits a O-labeling.

On the chromaticity of complete multipartite graphs with certain edges added

Discrete Mathematics, 2009

Let P(G, λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H, λ) = P(G, λ) implies H is isomorphic to G. For integers k ≥ 0, t ≥ 2, denote by K ((t − 1) × p, p + k) the complete t-partite graph that has t − 1 partite sets of size p and one partite set of size p + k. Let K(s, t, p, k) be the set of graphs obtained from K ((t − 1) × p, p + k) by adding a set S of s edges to the partite set of size p + k such that S is bipartite. If s = 1, denote the only graph in K(s, t, p, k) by K + ((t − 1) × p, p + k). In this paper, we shall prove that for k = 0, 1 and p + k ≥ s + 2, each graph G ∈ K (s, t, p, k) is chromatically unique if and only if S is a chromatically unique graph that has no cutvertex. As a direct consequence, the graph K + ((t − 1) × p, p + k) is chromatically unique for k = 0, 1 and p + k ≥ 3.

On graphs with complete multipartite -graphs

Discrete Mathematics, 2010

Jurišić and Koolen proposed to study 1-homogeneous distance-regular graphs, whose µ-graphs (that is, the graphs induced on the common neighbours of two vertices at distance 2) are complete multipartite. Examples include the Johnson graph J (8, 4), the halved 8-cube, the known generalized quadrangle of order (4, 2), an antipodal distance-regular graph constructed by T. Meixner and the Patterson graph. We investigate a more general situation, namely, requiring the graphs to have complete multipartite µ-graphs, and that the intersection number α exists, which means that for a triple (x, y, z) of vertices in Γ , such that x and y are adjacent and z is at distance 2 from x and y, the number α(x, y, z) of common neighbours of x, y and z does not depend on the choice of a triple. The latter condition is satisfied by any 1-homogeneous graph. Let K t×n denote the complete multipartite graph with t parts, each of which consists of an n-coclique. We show that if Γ is a graph whose µ-graphs are all isomorphic to K t×n and whose intersection number α exists, then α = t, as conjectured by Jurišić and Koolen, provided α ≥ 2. We also prove t ≤ 4, and that equality holds only when Γ is the unique distance-regular graph 3.O 7 (3).

On degree sets in k-partite graphs

Acta Universitatis Sapientiae, Informatica

The degree set of a k-partite graph is the set of distinct degrees of its vertices. We prove that every set of non-negative integers is a degree set of some k-partite graph.

On Cyclic Decompositions of Complete Graphs into Tripartite Graphs

Journal of Graph Theory, 2013

We introduce two new labelings for tripartite graphs and show that if a graph G with n edges admits either of these labelings, then there exists a cyclic G-decomposition of K 2nx+1 for every positive integer x. We also show that if G is the union of two vertext-disjoint cycles of odd length, other than C 3 ∪ C 3 , then G admits one of these labelings.

On the euclidean dimension of a complete multipartite graph

Discrete Mathematics, 1988

The euclidean dimension of a graph G, e(G), is the minimum n such that the vertices of G can be placed in euclidean n-space, R", in such a way that adjacent vertices have distance 1 and nonadjacent vertices have distances other than 1. Let G = K(n,,. , ns+,+J be a complete (s + t + u)-partite graph with vertex-classes consisting of s sets of size 1, I sets of size 2, and u sets of size 23. We prove that e(G)=s+t+2u if t+us2, and e(G)=s+t+2u-1 if t+uc1.

Bipartite Graphs and Monochromatic Squares

Order, 2018

Let κ be a successor cardinal. We prove that consistently every bipartite graph of size κ + × κ + contains either an independent set or a clique of size τ × τ for every ordinal τ < κ +. We prove a similar theorem for ℓ-partite graphs. 2010 Mathematics Subject Classification. 05C63.

On the cyclic decomposition of complete graphs into almost-bipartite graphs

Discrete Mathematics, 2004

Techniques of labeling the vertices of a bipartite graph G with n edges to yield cyclic G-decompositions of the complete graph K2nx+1 have received much attention in the literature. Up until recently, these techniques have been used mostly with bipartite graphs. An almost-bipartite graph is a non-bipartite graph with the property that the removal of a particular single edge renders the graph bipartite. Examples of such graphs include the odd cycles. Here we introduce the concept of a-labeling of an almost-bipartite graph and show that if an almost-bipartite graph G with n edges has a-labeling then there is a cyclic G-decomposition of K2nx+1 for all positive integers x. We also show that odd cycles as well as certain other almost-bipartite 2-regular graphs have-labelings.