A Degree-Preserving Builder--Chooser Game
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911718539526144 |
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| author | London, András |
| author_facet | London, András |
| contents | We propose a degree-preserving variant of the Builder--Chooser clique game of Pettie, Tardos, and Walczak. In each round, Builder chooses a matching, performs a degree-preserving growth (DPG) step by replacing the chosen edges with edges incident to a new vertex. Then partitions the entire edge set into two parts, and Chooser keeps one part. We begin the study of this game with the first nontrivial target, namely forcing a triangle. For triangle-free initial graphs we prove an exact one-round criterion, derive an exact one-round threshold on paths and exact forcing times on cycles, and identify the $5$-cycle as the first genuine two-round example. We then formulate a one-round criterion for larger cliques, prove a sharp exact result for forcing $K_4$ from triangle-free seeds. We establish general lower bounds on clique-forcing times from clique-free seeds, and isolate a conjectural template-amplifier lemma which, if proved, would imply that every clique is forceable from some triangle-free seed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26695 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Degree-Preserving Builder--Chooser Game London, András Combinatorics 05C57, 05C35, 05C55, 05C70 We propose a degree-preserving variant of the Builder--Chooser clique game of Pettie, Tardos, and Walczak. In each round, Builder chooses a matching, performs a degree-preserving growth (DPG) step by replacing the chosen edges with edges incident to a new vertex. Then partitions the entire edge set into two parts, and Chooser keeps one part. We begin the study of this game with the first nontrivial target, namely forcing a triangle. For triangle-free initial graphs we prove an exact one-round criterion, derive an exact one-round threshold on paths and exact forcing times on cycles, and identify the $5$-cycle as the first genuine two-round example. We then formulate a one-round criterion for larger cliques, prove a sharp exact result for forcing $K_4$ from triangle-free seeds. We establish general lower bounds on clique-forcing times from clique-free seeds, and isolate a conjectural template-amplifier lemma which, if proved, would imply that every clique is forceable from some triangle-free seed. |
| title | A Degree-Preserving Builder--Chooser Game |
| topic | Combinatorics 05C57, 05C35, 05C55, 05C70 |
| url | https://arxiv.org/abs/2605.26695 |