133 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
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
Around The Berge Problem And Hadwiger Conjecture
The Berge conjecture (see [1] or [2] or [3] or [5] or [6] or [7] or [9] or [10] or [11] ) was proved by Chudnovsky, Robertson, Seymour and Thomas in a paper of at least 140 pages (see [1]), and an elementary proof of the Berge conjecture was given by Ikorong Nemron in a detailled paper of 37 pages long (see [9]). The Hadwiger conjecture (see [4] or [5] or [7] or [8] or [10] or [11] or [12] or [13] or [15] or [16]) was proved by Ikorong Nemron in a detailled paper of 28 pages long (see [13]), by using arithmetic calculus, arithmetic congruences, elementary complex analysis, induction and reasoning by reduction to absurd. That being so, in this paper, via two simple Theorems,
we rigorously show that the difficult part of the Berge conjecture (solved) and the Hadwiger conjecture (also solved), are exactly the same conjecture. The previous immediately implies that, the Hadwiger conjecture is only a non obvious special case of the Berge conjecture
Defective and Clustered Graph Colouring
Consider the following two ways to colour the vertices of a graph where the
requirement that adjacent vertices get distinct colours is relaxed. A colouring
has "defect" if each monochromatic component has maximum degree at most
. A colouring has "clustering" if each monochromatic component has at
most vertices. This paper surveys research on these types of colourings,
where the first priority is to minimise the number of colours, with small
defect or small clustering as a secondary goal. List colouring variants are
also considered. The following graph classes are studied: outerplanar graphs,
planar graphs, graphs embeddable in surfaces, graphs with given maximum degree,
graphs with given maximum average degree, graphs excluding a given subgraph,
graphs with linear crossing number, linklessly or knotlessly embeddable graphs,
graphs with given Colin de Verdi\`ere parameter, graphs with given
circumference, graphs excluding a fixed graph as an immersion, graphs with
given thickness, graphs with given stack- or queue-number, graphs excluding
as a minor, graphs excluding as a minor, and graphs excluding
an arbitrary graph as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in
the Electronic Journal of Combinatoric
Connected matchings in special families of graphs.
A connected matching in a graph is a set of disjoint edges such that, for any pair of these edges, there is another edge of the graph incident to both of them. This dissertation investigates two problems related to finding large connected matchings in graphs. The first problem is motivated by a famous and still open conjecture made by Hadwiger stating that every k-chromatic graph contains a minor of the complete graph Kk . If true, Hadwiger\u27s conjecture would imply that every graph G has a minor of the complete graph K n/a(C), where a(G) denotes the independence number of G. For a graph G with a(G) = 2, Thomassé first noted the connection between connected matchings and large complete graph minors: there exists an ? \u3e 0 such that every graph G with a( G) = 2 contains K ?+, as a minor if and only if there exists a positive constant c such that every graph G with a( G) = 2 contains a connected matching of size cn. In Chapter 3 we prove several structural properties of a vertexminimal counterexample to these statements, extending work by Blasiak. We also prove the existence of large connected matchings in graphs with clique size close to the Ramsey bound by proving: for any positive constants band c with c \u3c ¼, there exists a positive integer N such that, if G is a graph with n =: N vertices, 0\u27( G) = 2, and clique size at most bv(n log(n) )then G contains a connected matching of size cn. The second problem concerns computational complexity of finding the size of a maximum connected matching in a graph. This problem has many applications including, when the underlying graph is chordal bipartite, applications to the bipartite margin shop problem. For general graphs, this problem is NP-complete. Cameron has shown the problem is polynomial-time solvable for chordal graphs. Inspired by this and applications to the margin shop problem, in Chapter 4 we focus on the class of chordal bipartite graphs and one of its subclasses, the convex bipartite graphs. We show that a polynomial-time algorithm to find the size of a maximum connected matching in a chordal bipartite graph reduces to finding a polynomial-time algorithm to recognize chordal bipartite graphs that have a perfect connected matching. We also prove that, in chordal bipartite graphs, a connected matching of size k is equivalent to several other statements about the graph and its biadjacency matrix, including for example, the statement that the complement of the latter contains a k x k submatrix that is permutation equivalent to strictly upper triangular matrix
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