Edge-spectral supersaturation for tripartite color-critical graphs

Abstract

We study edge-spectral supersaturation for two families of color-critical graphs with chromatic number three. For an integer $r\ge 1$, 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 $t+1\ge s\ge 3$) and $C_{2k+1}$ (when $r=k$), realized by split-graph constructions. First, let $t+1\ge s\ge 3$ 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 $\rho (G)>g_{s-1}(m)$ contains $\Omega (m^{(s+t-1)/2})$ copies of $K_{s,t}^{+}$. Second, for any fixed $k\ge 2$, the condition $\rho (G)>g_k(m)$ forces $N(C_{2k+1},G)=\Omega (m^k).$ 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.