The Merino--Welsh conjecture is false for matroids
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866916134902562816 |
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| author | Beke, Csongor Csáji, Gergely Kál Csikvári, Péter Pituk, Sára |
| author_facet | Beke, Csongor Csáji, Gergely Kál Csikvári, Péter Pituk, Sára |
| contents | The matroidal version of the Merino--Welsh conjecture states that the Tutte polynomial $T_M(x,y)$ of any matroid $M$ without loops and coloops satisfies that $$\max(T_M(2,0),T_M(0,2))\geq T_M(1,1).$$ Equivalently, if the Merino--Welsh conjecture is true for all matroids without loops and coloops, then the following inequalities are also satisfied for all matroids without loops and coloops: $$T_M(2,0)+T_M(0,2)\geq 2T_M(1,1),$$ and $$T_M(2,0)T_M(0,2)\geq T_M(1,1)^2.$$ We show a counter-example for these inequalities. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_01932 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Merino--Welsh conjecture is false for matroids Beke, Csongor Csáji, Gergely Kál Csikvári, Péter Pituk, Sára Combinatorics The matroidal version of the Merino--Welsh conjecture states that the Tutte polynomial $T_M(x,y)$ of any matroid $M$ without loops and coloops satisfies that $$\max(T_M(2,0),T_M(0,2))\geq T_M(1,1).$$ Equivalently, if the Merino--Welsh conjecture is true for all matroids without loops and coloops, then the following inequalities are also satisfied for all matroids without loops and coloops: $$T_M(2,0)+T_M(0,2)\geq 2T_M(1,1),$$ and $$T_M(2,0)T_M(0,2)\geq T_M(1,1)^2.$$ We show a counter-example for these inequalities. |
| title | The Merino--Welsh conjecture is false for matroids |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2311.01932 |