Abstract
We prove that for \(n>k\ge 3\), if G is an n-vertex graph with chromatic number k but any of its proper subgraphs has smaller chromatic number, then G contains at most \(n-k+3\) copies of a clique of size \(k-1\). This answers a problem of Abbott and Zhou and provides a tight bound on a conjecture of Gallai.
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Notes
An odd wheel is obtained from an odd cycle C by adding a new vertex x and joining x to every vertex of C. Note that an odd wheel on \(n\ge 6\) vertices is a 4-critical planar graph and has exactly \(n-1\) triangles.
Note that here we only use \(k-3\) incidence vectors from A to form \(\textbf{y}_j\)’s. In total, there are \(k-1\) elements of A that correspond to \(k-1\) colors. We have two special colors \(k-1\) and \(k-2\) set aside after Claim 1 and Claim 2, respectively, which leaves \(k-3\) colors.
If \(e\notin E(G)\), then it is evident that we have \(t(e,\mathcal {G})=0\).
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Acknowledgements
The authors would like to thank two referees and Stijn Cambie for their careful reading and for many valuable suggestions.
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J. Gao: Research supported by the Institute for Basic Science (IBS-R029-C4). J. Ma: Research supported by the National Key R and D Program of China 2020YFA0713100, National Natural Science Foundation of China grant 12125106, Innovation Program for Quantum Science and Technology 2021ZD0302902, and Anhui Initiative in Quantum Information Technologies Grant AHY150200.
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Gao, J., Ma, J. Tight Bounds Towards a Conjecture of Gallai. Combinatorica 43, 447–453 (2023). https://doi.org/10.1007/s00493-023-00020-z
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DOI: https://doi.org/10.1007/s00493-023-00020-z