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- W3198608204 abstract "A transversal of a partition is a set which contains exactly one element from each member of the partition and nothing else. A colouring of a graph is a partition of its vertex set into anticliques, that is, sets of pairwise nonadjacent vertices. We study the following problem: Given a transversal $T$ of a proper colouring $mathcal{C}$ of some graph $G$,is there a partition $mathfrak{H}$ of a subset of $V(G)$ into connected sets such that $T$ is a transversal of $mathfrak{H}$and any two distinct sets of $mathfrak{H}$ are adjacent?It has been conjectured by Matthias Kriesell that for any transversal $T$ of a colouring $mathcal{C}$ of order $k$ of some graph $G$ such that any pair of colour classes induces a connected subgraph, there exists such a partition $mathfrak{H}$ with pairwise adjacent sets. This would prove Hadwiger's conjecture for the class of uniquely optimally colourable graphs;however it is open for each $k geq 5$.This paper will provide an overview about the stated conjecture. It extracts associated results from my PhD thesis and the related papers, summarises their relevence to the stated problem, and discusses some unsuccessful attempts." @default.
- W3198608204 created "2021-09-13" @default.
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- W3198608204 date "2021-08-28" @default.
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- W3198608204 title "Rooted structures in graphs" @default.
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