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围长为6的混合笼的阶的新上界

New upper bounds on the order of mixed cages of girth 6

Gabriela Araujo-Pardo, Lydia Mirabel Mendoza-Cadena

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arXiv admin note: text overlap with arXiv:2503.17152
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AI总结：

本文研究围长为6的混合笼的阶，提出了一族无限混合图，在部分情况下改进了已有结果的上界。

摘要：

$[z,r;g]$-混合笼是指满足每个顶点有$z$个入弧、$z$个出弧、$r$条边且围长为$g$的最小阶混合图，其最小阶记为$n[z,r;g]$。本文提出了一族围长为6的无限混合图，在某些情况下改进了G. Araujo-Pardo和L. Mendoza-Cadena给出的族。特别地，若$q$为偶素数幂，我们构造的图族满足$n[\frac{q}{4},q;6]\leq 4q^2-4$；若$q$为奇素数幂且$\frac{q-4}{2}$为奇数，我们的图族满足$n[\frac{q-1}{4},q;6]\leq 4q^2-4$；否则满足$n[\frac{q-3}{4},q;6]\leq 4q^2-4$。

英文摘要：

A $[z,r;g]$-mixed cage is a mixed graph of minimum order such that each vertex has $z$ in-arcs, $z$ out-arcs, $r$ edges, and it has girth $g$, and the minimum order for $[z,r;g]$-mixed graphs is denoted by $n[z,r;g]$. In this paper, we present an infinite family of mixed graphs with girth $6$, that improves, in some cases, the families that we give in G. Araujo-Pardo and L. Mendoza-Cadena. \textit{On Mixed Cages of girth 6}, arXiv:2401.14768v2. In particular, if $q$ is an even prime power we construct a family of graphs that satisfies $n[\frac{q}{4},q;6]\leq 4q^2-4$, and if $q$ is an odd prime power, and $\frac{q-3}{2}$ is odd then our family satisfies that $n[\frac{q-1}{4},q;6]\leq 4q^2-4$, otherwise $n[\frac{q-3}{4},q;6]\leq 4q^2-4$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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