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具有高色指数的Steiner三元系

Steiner triple systems with high chromatic index

Darryn Bryant, Charles Colbourn, Daniel Horsley, Ian M. Wanless

备注：
10 pages, 0 figures
发表信息：
SIAM J. DISCRETE MATH. 31, (2017) 2603-2611
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摘要：

证明了每一个$v \neq 7$阶Steiner三元系的色指数至多为$(v+3)/2$，当$v \equiv 3 \pmod{6}$时，至多为$(v+5)/2$，当$v \equiv 1 \pmod{6}$时.本文构造了一个色指数至少为$(v+3)/2$的$v$阶的Steiner三元系，对于每个整数$v \equiv 3 \pmod{6}$，使得$v \geq 15$，除了四个可能的例外.我们进一步证明了在$v$中所构造的系统中不相交的并行类的最大数目是次线性的。最后，我们建立了每个阶$v \equiv 15 \pmod{18}$，至少有$v^{v^2(1/6+o(1))}$非同构Steiner三元系的色指数至少$(v+3)/2$，这些系统中的一些是循环的。

英文摘要：

It is conjectured that every Steiner triple system of order $v \neq 7$ has chromatic index at most $(v+3)/2$ when $v \equiv 3 \pmod{6}$ and at most $(v+5)/2$ when $v \equiv 1 \pmod{6}$. Herein, we construct a Steiner triple system of order $v$ with chromatic index at least $(v+3)/2$ for each integer $v \equiv 3 \pmod{6}$ such that $v \geq 15$, with four possible exceptions. We further show that the maximum number of disjoint parallel classes in the systems constructed is sublinear in $v$. Finally, we establish for each order $v \equiv 15 \pmod{18}$ that there are at least $v^{v^2(1/6+o(1))}$ non-isomorphic Steiner triple systems with chromatic index at least $(v+3)/2$ and that some of these systems are cyclic.

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