6 research outputs found
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
The structure of graphs not admitting a fixed immersion
We present an easy structure theorem for graphs which do not admit an immersion of the complete graph. The theorem motivates the definition of a variation of tree decompositions based on edge cuts instead of vertex cuts which we call tree-cut decompositions. We give a definition for the width of tree-cut decompositions, and using this definition along with the structure theorem for excluded clique immersions, we prove that every graph either has bounded tree-cut width or admits an immersion of a large wall
The structure of graphs not admitting a fixed immersion
We present an easy structure theorem for graphs which do not admit an
immersion of the complete graph. The theorem motivates the definition of a
variation of tree decompositions based on edge cuts instead of vertex cuts
which we call tree-cut decompositions. We give a definition for the width of
tree-cut decompositions, and using this definition along with the structure
theorem for excluded clique immersions, we prove that every graph either has
bounded tree-cut width or admits an immersion of a large wall
Forbidding Kuratowski Graphs as Immersions
The immersion relation is a partial ordering relation on graphs that is
weaker than the topological minor relation in the sense that if a graph
contains a graph as a topological minor, then it also contains it as an
immersion but not vice versa. Kuratowski graphs, namely and ,
give a precise characterization of planar graphs when excluded as topological
minors. In this note we give a structural characterization of the graphs that
exclude Kuratowski graphs as immersions. We prove that they can be constructed
by applying consecutive -edge-sums, for , starting from graphs that
are planar sub-cubic or of branch-width at most 10
Forbidding Kuratowski graphs as immersions
The immersion relation is a partial ordering relation on graphs that is weaker than the topological minor relation in the sense that if a graph G contains a graph H as a topological minor, then it also contains it as an immersion but not vice versa. Kuratowski graphs, namely K 5 and K 3,3 , give a precise characterization of planar graphs when excluded as topological minors. In this note we give a structural characterization of the graphs that exclude Kuratowski graphs as immersions. We prove that they can be constructed by applying consecutive i-edge-sums, for i ≤ 3, starting from graphs that are planar sub-cubic or of branchwidth at most 10