7 research outputs found
A note on forbidding clique immersions
Robertson and Seymour proved that the relation of graph immersion is
well-quasi-ordered for finite graphs. Their proof uses the results of graph
minors theory. Surprisingly, there is a very short proof of the corresponding
rough structure theorem for graphs without -immersions; it is based on the
Gomory-Hu theorem. The same proof also works to establish a rough structure
theorem for Eulerian digraphs without -immersions, where
denotes the bidirected complete digraph of order
Constructing graphs with no immersion of large complete graphs
In 1989, Lescure and Meyniel proved, for , that every -chromatic
graph contains an immersion of , and in 2003 Abu-Khzam and Langston
conjectured that this holds for all . In 2010, DeVos, Kawarabayashi, Mohar,
and Okamura proved this conjecture for . In each proof, the
-chromatic assumption was not fully utilized, as the proofs only use the
fact that a -critical graph has minimum degree at least . DeVos,
Dvo\v{r}\'ak, Fox, McDonald, Mohar, and Scheide show the stronger conjecture
that a graph with minimum degree has an immersion of fails for
and with a finite number of examples for each value of ,
and small chromatic number relative to , but it is shown that a minimum
degree of does guarantee an immersion of .
In this paper we show that the stronger conjecture is false for
and give infinite families of examples with minimum degree and chromatic
number or that do not contain an immersion of . Our examples
can be up to -edge-connected. We show, using Haj\'os' Construction, that
there is an infinite class of non--colorable graphs that contain an
immersion of . We conclude with some open questions, and the conjecture
that a graph with minimum degree and more than
vertices of degree at least has an immersion of
Immersion of complete digraphs in Eulerian digraphs
A digraph \emph{immerses} a digraph if there is an injection and a collection of pairwise edge-disjoint directed paths
, for , such that starts at and ends at .
We prove that every Eulerian digraph with minimum out-degree immerses a
complete digraph on vertices, thus answering a question of DeVos,
Mcdonald, Mohar, and Scheide.Comment: 17 pages; fixed typo
Sublinear expanders and their applications
In this survey we aim to give a comprehensive overview of results using
sublinear expanders. The term sublinear expanders refers to a variety of
definitions of expanders, which typically are defined to be graphs such
that every not-too-small and not-too-large set of vertices has
neighbourhood of size at least , where is a function of
and . This is in contrast with linear expanders, where is
typically a constant. :We will briefly describe proof ideas of some of the
results mentioned here, as well as related open problems.Comment: 39 pages, 15 figures. This survey will appear in `Surveys in
Combinatorics 2024' (the proceedings of the 30th British Combinatorial
Conference