AI 中文总结
研究无钻石无爪立方图的强边着色问题，通过展示一个18个顶点的立方图\(H\)，其\(\chi'_s(H)=7\)，证明在简单立方图中相关猜想不成立，还表明两个关于强边着色的表述不等价，特定版本问题仍未解决。
AI 中文摘要
强边着色是一种恰当的边着色，其中每个颜色类都是诱导匹配；颜色的最小数量是强色指数\(\chi'_s(G)\)。林和林证明，除三棱柱外的每个无爪次立方图都满足\(\chi'_s(G) \leq 7\)，其所有紧密例子都包含钻石。卡多斯问，每个无钻石无爪立方图是否都能强6边着色，等价于对于每个立方图\(G\)，\(\chi'_s(T(G)) = 6\)是否成立，其中\(T(G)\)是\(G\)的截断。我们展示了一个明确的、连通的、简单的、无钻石无爪的18个顶点的立方图\(H\)，其\(\chi'_s(H) = 7\)，并表明它是这种非棱柱例子中顶点最少的。因此，在简单立方图中，即使排除棱柱，第一个表述也是错误的；并且由于\(H\)是一个有平行边的立方多重图的截断，除非“立方图”允许表示无环多重图，否则这两个表述不等价，在这种理解下问题的答案是否定的。仅限于简单基图截断的更窄版本仍然开放。
英文摘要
A strong edge coloring is a proper edge coloring in which every color class is an induced matching; the least number of colors is the strong chromatic index $χ'_s(G)$. Lin and Lin proved that every claw-free subcubic graph other than the triangular prism satisfies $χ'_s(G) \le 7$, with all their tight examples containing diamonds. Kardos (Problem 4.1 of the open-problem collection of the 33rd Workshop on Cycles and Colourings) asked whether every diamond-free claw-free cubic graph is strongly 6-edge-colorable, equivalently whether $χ'_s(T(G))=6$ for every cubic graph $G$, where $T(G)$ is the truncation of $G$. We exhibit an explicit connected, simple, diamond-free, claw-free cubic graph $H$ on 18 vertices with $χ'_s(H)=7$, and show that it has the fewest vertices possible for such a non-prism example. Hence the first formulation, over simple cubic graphs, is false even after excluding the prism; and since $H$ is the truncation of a cubic multigraph with parallel edges, the two formulations are not equivalent unless "cubic graph" is allowed to mean loopless multigraph, under which reading the problem is answered negatively. The narrower version restricted to truncations of simple base graphs remains open.
Comments7 pages, 2 figures; verification script included as an ancillary file
