小团数与大的最小度图的有界色数

中文摘要 AI 辅助

我们证明每个最小度至少为$\frac{n}{3}$的无三角形图都是$4$-可染色的，从而解决了Brandt和Thomassé（2005）在阈值$\frac{n}{3}$处提出的问题。数字4是最优的。对于正整数值函数$f(n)=o(n)$，我们将Kneser图$KG(n,f(n))$的$f(n)$顶点子图的色数与最小度至少为$\frac{n}{3}-f(n)$的无三角形图的色数联系起来。因此，对于每个$0<\delta<1$和$\varepsilon>0$，以及所有足够大的$n$，每个最小度至少为$\frac{n}{3}-n^{1-\delta}$的$n$顶点无三角形图的色数至多为$10^{391}+1+\left\lceil{(1+\varepsilon)(1-\delta)}/{\delta}\right\rceil$。我们还证明，每个足够大的、最小度至少为$\frac{n}{3}-f(n)$且色数至少为$10^{391}$的$n$顶点极大无三角形图包含一个二部子图，其两部分的阶分别为$\frac{n}{3}-O(f(n))$和$\frac{2n}{3}-O(f(n))$；剩余的诱导子图允许同态到$KG(\frac{n}{3}-O(f(n)),O(f(n)))$。最后，我们将最小度至少为$\frac{2r-5}{2r-3}n-f(n)$的极大$K_r$-自由图与$K_{r-1}$-自由图联系起来，并将这些结果扩展到$K_r$-自由图。我们的证明采用了Łuczak、Polcyn和Reiher最近的强Brandt--Thomassé定理。

英文摘要

We prove that every triangle-free graph with minimum degree at least $\frac{n}{3}$ is $4$-colorable and thereby settle a problem of Brandt and Thomassé (2005) at the threshold $\frac{n}{3}$. The number four is best possible. For a positive integer-valued function $f(n)=o(n)$, we relate the chromatic number of $f(n)$-vertex subgraphs of the Kneser graph $KG(n,f(n))$ to that of triangle-free graphs with minimum degree at least $\frac{n}{3}-f(n)$. Consequently, for every $0<δ<1$ and $\varepsilon>0$, and for all sufficiently large $n$, every $n$-vertex triangle-free graph with minimum degree at least $\frac{n}{3}-n^{1-δ}$ has chromatic number at most $10^{391}+1+\left\lceil{(1+\varepsilon)(1-δ)}/δ\right\rceil$. We also show that every sufficiently large $n$-vertex maximal triangle-free graph with minimum degree at least $\frac{n}{3}-f(n)$ and chromatic number at least $10^{391}$ contains a bipartite subgraph with parts of orders $\frac{n}{3}-O(f(n))$ and $\frac{2n}{3}-O(f(n))$; the remaining induced subgraph admits a homomorphism to $KG(\frac{n}{3}-O(f(n)),O(f(n)))$. Finally, we connect maximal $K_r$-free graphs with minimum degree at least $\frac{2r-5}{2r-3}n-f(n)$ to $K_{r-1}$-free graphs and extend these results to $K_r$-free graphs. Our proofs employ the recent strong Brandt--Thomassé theorem of Łuczak, Polcyn, and Reiher.

发表机构

• School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院和LPMC）

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