Abstract
We determine the maximum number of edges that a chordal graph G can have if its degree, \(\Delta (G)\), and its matching number, \(\nu (G)\), are bounded. To do so, we show that for every \(d,\nu \in \mathbb {N}\), there exists a chordal graph G with \(\Delta (G)<d\) and \(\nu (G)<\nu \) whose number of edges matches the upper bound, while having a simple structure: it is a disjoint union of cliques and stars.
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Notes
- 1.
Statements marked with \(\spadesuit \) had their proofs omitted due to space constraints.
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Blair, J.R.S., Heggernes, P., Lima, P.T., Lokshtanov, D. (2020). On the Maximum Number of Edges in Chordal Graphs of Bounded Degree and Matching Number. In: Kohayakawa, Y., Miyazawa, F.K. (eds) LATIN 2020: Theoretical Informatics. LATIN 2021. Lecture Notes in Computer Science(), vol 12118. Springer, Cham. https://doi.org/10.1007/978-3-030-61792-9_47
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