545 research outputs found
The list-chromatic index of K 6
We prove that the list-chromatic index and paintability index of K"6 is 5. That indeed @g"@?^'(K"6)=5 was a still open special case of the List Coloring Conjecture. Our proof demonstrates how colorability problems can numerically be approached by the use of computer algebra systems and the Combinatorial Nullstellensatz
The List Square Coloring Conjecture fails for cubic bipartite graphs and planar line graphs
Kostochka and Woodall (2001) conjectured that the square of every graph has
the same chromatic number and list chromatic number. In 2015 Kim and Park
disproved this conjecture for non-bipartie graphs and alternatively they
developed their construction to bipartite graphs such that one partite set has
maximum degree . Motivated by the List Total Coloring Conjecture, they also
asked whether this number can be pushed down to . At about the same time,
Kim, SooKwon, and Park (2015) asked whether there would exist a claw-free
counterexample to establish a generalization for a conjecture of Gravier and
Maffray (1997). In this note, we answer the problem of Kim and Park by pushing
the desired upper bound down to by introducing a family of cubic bipartite
counterexamples, and positively answer the problem of Kim, SooKwon, and Park by
introducing a family of planar line graphs
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
Graphs and graph polynomials
A dissertation submitted to the School of Mathematics in fulfilment of the requirements for the degree of Master of Science School of Mathematics University of the Witwatersrand, October 2017In this work we study the k-defect polynomials of a graph G. The k defect polynomial
is a function in λ that gives the number of improper colourings of a graph using
λ colours. The k-defect polynomials generate the bad colouring polynomial which
is equivalent to the Tutte polynomial, hence their importance in a more general
graph theoretic setting. By setting up a one-to-one correspondence between triangular
numbers and complete graphs, we use number theoretical methods to study certain
characteristics of the k-defect polynomials of complete graphs. Specifically we are able
to generate an expression for any k-defect polynomial of a complete graph, determine
integer intervals for k on which the k-defect polynomials for complete graphs are equal
to zero and also determine a formula to calculate the minimum number of k-defect
polynomials that are equal to zero for any complete graph.XL201
On topological relaxations of chromatic conjectures
There are several famous unsolved conjectures about the chromatic number that
were relaxed and already proven to hold for the fractional chromatic number. We
discuss similar relaxations for the topological lower bound(s) of the chromatic
number. In particular, we prove that such a relaxed version is true for the
Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of
Hadwiger from this point of view. For the latter, a similar statement was
already proven in an earlier paper of the first author with G. Tardos, our main
concern here is that the so-called odd Hadwiger conjecture looks much more
difficult in this respect. We prove that the statement of the odd Hadwiger
conjecture holds for large enough Kneser graphs and Schrijver graphs of any
fixed chromatic number
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