16 research outputs found
Average degree conditions forcing a minor
Mader first proved that high average degree forces a given graph as a minor.
Often motivated by Hadwiger's Conjecture, much research has focused on the
average degree required to force a complete graph as a minor. Subsequently,
various authors have consider the average degree required to force an arbitrary
graph as a minor. Here, we strengthen (under certain conditions) a recent
result by Reed and Wood, giving better bounds on the average degree required to
force an -minor when is a sparse graph with many high degree vertices.
This solves an open problem of Reed and Wood, and also generalises (to within a
constant factor) known results when is an unbalanced complete bipartite
graph
Densities of Minor-Closed Graph Families
We define the limiting density of a minor-closed family of simple graphs F to
be the smallest number k such that every n-vertex graph in F has at most
kn(1+o(1)) edges, and we investigate the set of numbers that can be limiting
densities. This set of numbers is countable, well-ordered, and closed; its
order type is at least {\omega}^{\omega}. It is the closure of the set of
densities of density-minimal graphs, graphs for which no minor has a greater
ratio of edges to vertices. By analyzing density-minimal graphs of low
densities, we find all limiting densities up to the first two cluster points of
the set of limiting densities, 1 and 3/2. For multigraphs, the only possible
limiting densities are the integers and the superparticular ratios i/(i+1).Comment: 19 pages, 4 figure
Forcing a sparse minor
This paper addresses the following question for a given graph : what is
the minimum number such that every graph with average degree at least
contains as a minor? Due to connections with Hadwiger's Conjecture,
this question has been studied in depth when is a complete graph. Kostochka
and Thomason independently proved that . More generally,
Myers and Thomason determined when has a super-linear number of
edges. We focus on the case when has a linear number of edges. Our main
result, which complements the result of Myers and Thomason, states that if
has vertices and average degree at least some absolute constant, then
. Furthermore, motivated by the case when
has small average degree, we prove that if has vertices and edges,
then (where the coefficient of 1 in the term is best
possible)
Maximum spread of -minor-free graphs
The spread of a graph is the difference between the largest and smallest
eigenvalues of the adjacency matrix of . In this paper, we consider the
family of graphs which contain no -minor. We show that for any , there is an integer such that the maximum spread of an -vertex
-minor-free graph is achieved by the graph obtained by joining a
vertex to the disjoint union of copies of
and isolated vertices. The
extremal graph is unique, except when and is an integer, in which case the other extremal graph is the graph
obtained by joining a vertex to the disjoint union of copies of and isolated vertices. Furthermore, we give an
explicit formula for .Comment: 15 pages. arXiv admin note: text overlap with arXiv:2209.1377
On the choosability of -minor-free graphs
Given a graph , let us denote by and ,
respectively, the maximum chromatic number and the maximum list chromatic
number of -minor-free graphs. Hadwiger's famous coloring conjecture from
1943 states that for every . In contrast, for list
coloring it is known that
and thus, is bounded away from the conjectured value for
by at least a constant factor. The so-called -Hadwiger's
conjecture, proposed by Seymour, asks to prove that
for a given graph (which would be implied by Hadwiger's conjecture). In
this paper, we prove several new lower bounds on , thus exploring
the limits of a list coloring extension of -Hadwiger's conjecture. Our main
results are:
For every and all sufficiently large graphs we have
, where
denotes the vertex-connectivity of .
For every there exists such that
asymptotically almost every -vertex graph with edges satisfies .
The first result generalizes recent results on complete and complete
bipartite graphs and shows that the list chromatic number of -minor-free
graphs is separated from the natural lower bound by a
constant factor for all large graphs of linear connectivity. The second
result tells us that even when is a very sparse graph (with an average
degree just logarithmic in its order), can still be separated from
by a constant factor arbitrarily close to . Conceptually
these results indicate that the graphs for which is close to
are typically rather sparse.Comment: 14 page
Proper conflict-free list-coloring, odd minors, subdivisions, and layered treewidth
Proper conflict-free coloring is an intermediate notion between proper
coloring of a graph and proper coloring of its square. It is a proper coloring
such that for every non-isolated vertex, there exists a color appearing exactly
once in its (open) neighborhood. Typical examples of graphs with large proper
conflict-free chromatic number include graphs with large chromatic number and
bipartite graphs isomorphic to the -subdivision of graphs with large
chromatic number. In this paper, we prove two rough converse statements that
hold even in the list-coloring setting. The first is for sparse graphs: for
every graph , there exists an integer such that every graph with no
subdivision of is (properly) conflict-free -choosable. The second
applies to dense graphs: every graph with large conflict-free choice number
either contains a large complete graph as an odd minor or contains a bipartite
induced subgraph that has large conflict-free choice number. These give two
incomparable (partial) answers of a question of Caro, Petru\v{s}evski and
\v{S}krekovski. We also prove quantitatively better bounds for minor-closed
families, implying some known results about proper conflict-free coloring and
odd coloring in the literature. Moreover, we prove that every graph with
layered treewidth at most is (properly) conflict-free -choosable.
This result applies to -planar graphs, which are graphs whose coloring
problems have attracted attention recently.Comment: Hickingbotham recently independently announced a paper
(arXiv:2203.10402) proving a result similar to the ones in this paper. Please
see the notes at the end of this paper for details. v2: add results for odd
minors, which applies to graphs with unbounded degeneracy, and change the
title of the pape