1,443 research outputs found
Ramsey numbers in complete balanced multipartite graphs. Part I: Set numbers
AbstractThe notion of a graph theoretic Ramsey number is generalised by assuming that both the original graph whose edges are arbitrarily bi-coloured and the sought after monochromatic subgraphs are complete, balanced, multipartite graphs, instead of complete graphs as in the classical definition. We previously confined our attention to diagonal multipartite Ramsey numbers. In this paper the definition of a multipartite Ramsey number is broadened still further, by incorporating off-diagonal numbers, fixing the number of vertices per partite set in the larger graph and then seeking the minimum number of such partite sets that would ensure the occurrence of certain specified monochromatic multipartite subgraphs
Graph Saturation in Multipartite Graphs
Let be a fixed graph and let be a family of graphs. A
subgraph of is -saturated if no member of
is a subgraph of , but for any edge in , some element of
is a subgraph of . We let and
denote the maximum and minimum size of an
-saturated subgraph of , respectively. If no element of
is a subgraph of , then .
In this paper, for and we determine
, where is the complete balanced -partite
graph with partite sets of size . We also give several families of
constructions of -saturated subgraphs of for . Our results
and constructions provide an informative contrast to recent results on the
edge-density version of from [A. Bondy, J. Shen, S.
Thomass\'e, and C. Thomassen, Density conditions for triangles in multipartite
graphs, Combinatorica 26 (2006), 121--131] and [F. Pfender, Complete subgraphs
in multipartite graphs, Combinatorica 32 (2012), no. 4, 483--495].Comment: 16 pages, 4 figure
Transversal designs and induced decompositions of graphs
We prove that for every complete multipartite graph there exist very
dense graphs on vertices, namely with as many as
edges for all , for some constant , such that can be
decomposed into edge-disjoint induced subgraphs isomorphic to~. This result
identifies and structurally explains a gap between the growth rates and
on the minimum number of non-edges in graphs admitting an
induced -decomposition
A Short Note on Undirected Fitch Graphs
The symmetric version of Fitch's xenology relation coincides with class of
complete multipartite graph and thus cannot convey any non-trivial phylogenetic
information
Induced minors and well-quasi-ordering
A graph is an induced minor of a graph if it can be obtained from an
induced subgraph of by contracting edges. Otherwise, is said to be
-induced minor-free. Robin Thomas showed that -induced minor-free
graphs are well-quasi-ordered by induced minors [Graphs without and
well-quasi-ordering, Journal of Combinatorial Theory, Series B, 38(3):240 --
247, 1985].
We provide a dichotomy theorem for -induced minor-free graphs and show
that the class of -induced minor-free graphs is well-quasi-ordered by the
induced minor relation if and only if is an induced minor of the gem (the
path on 4 vertices plus a dominating vertex) or of the graph obtained by adding
a vertex of degree 2 to the complete graph on 4 vertices. To this end we proved
two decomposition theorems which are of independent interest.
Similar dichotomy results were previously given for subgraphs by Guoli Ding
in [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489--502,
1992] and for induced subgraphs by Peter Damaschke in [Induced subgraphs and
well-quasi-ordering, Journal of Graph Theory, 14(4):427--435, 1990]
Finite -connected homogeneous graphs
A finite graph \G is said to be {\em -connected homogeneous}
if every isomorphism between any two isomorphic (connected) subgraphs of order
at most extends to an automorphism of the graph, where is a
group of automorphisms of the graph. In 1985, Cameron and Macpherson determined
all finite -homogeneous graphs. In this paper, we develop a method for
characterising -connected homogeneous graphs. It is shown that for a
finite -connected homogeneous graph \G=(V, E), either G_v^{\G(v)} is
--transitive or G_v^{\G(v)} is of rank and \G has girth , and
that the class of finite -connected homogeneous graphs is closed under
taking normal quotients. This leads us to study graphs where is
quasiprimitive on . We determine the possible quasiprimitive types for
in this case and give new constructions of examples for some possible types
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