三部染色临界图的边谱过饱和
Edge-spectral supersaturation for tripartite color-critical graphs
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中文总结 AI 辅助
该研究针对色数为3的两类染色临界图，证明超过紧谱Turán阈值会迫使出现最优多项式数量的对应图副本，扩展了边谱存在性定理至三色过饱和结果。
中文摘要 AI 辅助
我们研究两类色数为3的染色临界图的边谱过饱和问题。对于整数$r\geq1$，定义谱阈值$g_r(m):=\frac{r-1+\sqrt{4m-r^2+1}}{2}$，这是避开$K_{s,t}^+$（当$t+1\geq s\geq3$时）和$C_{2k+1}$（当$r=k$时）的图的谱半径的紧上界，由分裂图构造实现。首先，设$t+1\geq s\geq3$为固定整数，$K_{s,t}^+$是在$K_{s,t}$的大小为$s$的部分添加一条边得到的图，我们证明：每个足够大的$m$边图$G$，若其谱半径$\rho(G)>g_{s-1}(m)$，则包含$\Omega(m^{(s+t-1)/2})$个$K_{s,t}^+$副本。其次，对于任意固定的$k\geq2$，条件$\rho(G)>g_k(m)$会迫使$N(C_{2k+1},G)=\Omega(m^k)$。我们还构造图表明，这两个下界在常数因子范围内是紧的。这些结果表明，超过紧谱Turán阈值$g_r(m)$不仅会迫使出现单个副本，还会迫使出现这些染色临界图的最优多项式数量的副本，因此，超过相关分裂图谱阈值会迫使出现最优多项式阶数的副本，将边谱存在性定理扩展到微妙的三色 regime（区域）的过饱和结果中。
英文摘要
We study edge-spectral supersaturation for two families of color-critical graphs with chromatic number three. For an integer $r\geq 1$, we define the spectral threshold \[ g_r(m):=\frac{r-1+\sqrt{4m-r^2+1}}{2}, \] which is the tight upper bound on the spectral radius of graphs avoiding $K_{s,t}^+$ (when $t+1\geq s\geq 3$) and $C_{2k+1}$ (when $r=k$), realized by split-graph constructions. First, let $t+1 \geq s\geq 3$ be fixed integers, and let $K_{s,t}^{+}$ be obtained by adding an edge to the part of size $s$ in $K_{s,t}$. We prove that every sufficiently large $m$-edge graph $G$ with $ρ(G)>g_{s-1}(m)$ contains $Ω(m^{(s+t-1)/2})$ copies of $K_{s,t}^{+}$. Second, for any fixed $k\geq 2$, the condition $ρ(G)>g_k(m)$ forces $N(C_{2k+1},G)=Ω(m^k).$ We also construct graphs showing that both lower bounds are tight up to constant factors. These results establish that exceeding the tight spectral Turán threshold $g_r(m)$ forces not just a single copy, but the optimal polynomial number of copies of these color-critical graphs. Thus, crossing the relevant split-graph spectral threshold forces the optimal polynomial order of copies, extending edge-spectral existence theorems to supersaturation results in the delicate three-chromatic regime.
