发表机构
Wuchang Shouyi College（武昌首义学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在Cranston研究基础上，通过Kempe链构造等方法，证明了Albertson猜想在r=27、28、29时成立，填补了相关剩余情况的研究空白。
AI 中文摘要
Albertson猜想指出，每个满足色数χ(G)≥r的有限简单图G，其交叉数cr(G)至少等于完全图K_r的交叉数cr(K_r)。在Cranston已验证r≤24的情况，并将r∈{25,26}的情况归约为三个剩余阶数的基础上，本文排除了这些剩余情况，进而证明了r=27、28、29的情况。第一个结构要素是Kempe链构造：若k-临界图存在一个度数为k-1的顶点，则它包含K_k的分支干净本质浸入。本质浸入具有交叉单调性，因此临界反例的最小度数至少为k。对于r=27，这一单位度数增益、Gallai的联结构造、临界图边数界以及诱导子图平均法可封闭所有可能的阶数。对于r=28和r=29，剩余的接近2r的阶数被转化为稠密补图。Stehlík的着色定理使奇阶补图成为因子临界图；团划分障碍产生反紧匹配性质；而Tutte障碍、Hall型扩张以及亏缺记账法排除了最终情况。对于r=29的阶数58，Rabern的着色不等式处理正则情况，最后一个度数亏缺为2的情况被归约为两个不相交三角形和有限障碍分析。
英文摘要
Albertson's conjecture asserts that every finite simple graph $G$ with $χ(G) \ge r$ satisfies $\operatorname{cr}(G) \ge \operatorname{cr}(K_r)$. Building on Cranston's verification for $r \le 24$ and his reduction of $r \in \{25,26\}$ to three residual orders, we eliminate those residual cases and then prove the cases $r=27,28,29$. The first structural ingredient is a Kempe-chain construction: if a $k$-critical graph has a vertex of degree $k-1$, then it contains a branch-clean essential immersion of $K_k$. Essential immersions are crossing-monotone, so a critical counterexample must have minimum degree at least $k$. For $r=27$, this one-unit degree gain, Gallai's join structure, critical-graph edge bounds, and induced-subgraph averaging close every possible order. For $r=28$ and $r=29$, the remaining near-$2r$ orders are converted to dense complements. Stehlík's coloring theorem makes the odd-order complements factor-critical; a clique-partition obstruction yields an anti-tight matching property; and Tutte barriers, Hall-type expansion, and deficit bookkeeping eliminate the final cases. At order 58 for $r=29$, Rabern's coloring inequality handles the regular case, while the last degree-deficit-two case is reduced to two disjoint triangles and a finite barrier analysis.
Comments23 pages; ancillary files include exact-arithmetic and finite-case verification materials
