Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Colouring Graphs of Bounded Diameter in the Absence of Small Cycles   For $k\geq 1$, a $k$-colouring $c$ of $G$ is a mapping from $V(G)$ to
$\{1,2,\ldots,k\}$ such that $c(u)\neq c(v)$ for any two non-adjacent vertices
$u$ and $v$. The $k$-Colouring problem is to decide if a graph $G$ has a
$k$-colouring. For a family of graphs ${\cal H}$, a graph $G$ is ${\cal
H}$-free if $G$ does not contain any graph from ${\cal H}$ as an induced
subgraph. Let $C_s$ be the $s$-vertex cycle. In previous work (MFCS 2019) we
examined the effect of bounding the diameter on the complexity of $3$-Colouring
for $(C_3,\ldots,C_s)$-free graphs and $H$-free graphs where $H$ is some
polyad. Here, we prove for certain small values of $s$ that $3$-Colouring is
polynomial-time solvable for $C_s$-free graphs of diameter $2$ and
$(C_4,C_s)$-free graphs of diameter $2$. In fact, our results hold for the more
general problem List $3$-Colouring. We complement these results with some
hardness result for diameter $4$.

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The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
14
The categories of this paper are: math-ph
22
The categories of this paper are: chao-dyn
0
The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
18
The categories of this paper are: econ
10
The categories of this paper are: q-bio
28
The categories of this paper are: alg-geom
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The categories of this paper are: physics
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The categories of this paper are: hep-th
37
The categories of this paper are: supr-con
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The categories of this paper are: acc-phys
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The categories of this paper are: bayes-an
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin