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
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)
On \u3cem\u3eK\u3c/em\u3e\u3cem\u3e\u3csub\u3es,t\u3c/sub\u3e\u3c/em\u3e-minors in Graphs with Given Average Degree
Let D(H) be the minimum d such that every graph G with average degree d has an H-minor. Myers and Thomason found good bounds on D(H) for almost all graphs H and proved that for \u27balanced\u27 H random graphs provide extremal examples and determine the extremal function. Examples of \u27unbalanced graphs\u27 are complete bipartite graphs Ks,t for a fixed s and large t. Myers proved upper bounds on D(Ks,t ) and made a conjecture on the order of magnitude of D(Ks,t ) for a fixed s and t → ∞. He also found exact values for D(K2,t) for an infinite series of t. In this paper, we confirm the conjecture of Myers and find asymptotically (in s) exact bounds on D(Ks,t ) for a fixed s and large t
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
Subdivisions in a bipartite graph
Given a bipartite graph G with m and n vertices, respectively,in its vertices classes, and given two integers s, t such that 2 ≤ s ≤ t, 0 ≤ m−s ≤ n−t, and m+n ≤ 2s+t−1, we prove that if G has at least mn−(2(m−s)+n−t) edges then it contains a subdivision of the complete bipartite with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn − (2(m − s) + n − t + 1) edges for this topological Turan type problem.Peer Reviewe
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
Dense graphs have K3,t minors
AbstractLet K3,t∗ denote the graph obtained from K3,t by adding all edges between the three vertices of degree t in it. We prove that for each t≥6300 and n≥t+3, each n-vertex graph G with e(G)>12(t+3)(n−2)+1 has a K3,t∗-minor. The bound is sharp in the sense that for every t, there are infinitely many graphs G with e(G)=12(t+3)(|V(G)|−2)+1 that have no K3,t-minor. The result confirms a partial case of the conjecture by Woodall and Seymour that every (s+t)-chromatic graph has a Ks,t-minor