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تسجيل مستخدم جديدColouring Graphs of Bounded Diameter in the Absence of Small Cycles
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For $kgeq 1$, a $k$-colouring $c$ of $G$ is a mapping from $V(G)$ to ${1,2,ldots,k}$ such that $c(u) eq c(v)$ 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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