11,142 research outputs found

### Independent Sets in Graphs with an Excluded Clique Minor

Let $G$ be a graph with $n$ vertices, with independence number $\alpha$, and
with with no $K_{t+1}$-minor for some $t\geq5$. It is proved that
$(2\alpha-1)(2t-5)\geq2n-5$

### On Tree-Partition-Width

A \emph{tree-partition} of a graph $G$ is a proper partition of its vertex
set into `bags', such that identifying the vertices in each bag produces a
forest. The \emph{tree-partition-width} of $G$ is the minimum number of
vertices in a bag in a tree-partition of $G$. An anonymous referee of the paper
by Ding and Oporowski [\emph{J. Graph Theory}, 1995] proved that every graph
with tree-width $k\geq3$ and maximum degree $\Delta\geq1$ has
tree-partition-width at most $24k\Delta$. We prove that this bound is within a
constant factor of optimal. In particular, for all $k\geq3$ and for all
sufficiently large $\Delta$, we construct a graph with tree-width $k$, maximum
degree $\Delta$, and tree-partition-width at least (\eighth-\epsilon)k\Delta.
Moreover, we slightly improve the upper bound to ${5/2}(k+1)({7/2}\Delta-1)$
without the restriction that $k\geq3$

### Drawing a Graph in a Hypercube

A $d$-dimensional hypercube drawing of a graph represents the vertices by
distinct points in $\{0,1\}^d$, such that the line-segments representing the
edges do not cross. We study lower and upper bounds on the minimum number of
dimensions in hypercube drawing of a given graph. This parameter turns out to
be related to Sidon sets and antimagic injections.Comment: Submitte

### Colouring the Square of the Cartesian Product of Trees

We prove upper and lower bounds on the chromatic number of the square of the
cartesian product of trees. The bounds are equal if each tree has even maximum
degree

### 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" $d$ if each monochromatic component has maximum degree at most
$d$. A colouring has "clustering" $c$ if each monochromatic component has at
most $c$ 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
$K_t$ as a minor, graphs excluding $K_{s,t}$ as a minor, and graphs excluding
an arbitrary graph $H$ 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

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