Unbalanced Tur\'an and spectral Tur\'an problems with prescribed large maximum degree

Abstract

Classical Tur\'an theory shows that, without additional constraints, the extremal structure of $F$-free graphs is governed by the chromatic number of $F$ and is asymptotically the balanced Tur\'an graph. We investigate how prescribing a large maximum degree changes this picture and leads to an unbalanced Tur\'an problem. Let $F$ be a graph with $\chi(F)=r+1\ge3$, and let $\lceil(r-1)n/r\rceil\le\Delta\le n-1$. We study the maximum number of edges and adjacency spectral radius of an $n$-vertex $F$-free graph with maximum degree exactly $\Delta$. For $F=K_{r+1}$, the unbalanced $r$-partite graph $S_{n,\Delta}^{(r)}=(n-\Delta)K_1\vee T(\Delta,r-1)$ is the unique maximizer of both quantities. Let $a(F)$ be the minimum size of an independent set $I$ such that $\chi(F-I)\le r$. If $a(F)=1$, we prove edge and spectral stability with respect to $S_{n,\Delta}^{(r)}$. If $a(F)>1$, the extremal values are $t(n,r)+o(n^2)$ and $\rho(T(n,r))+o(n)$, respectively, and every extremal graph differs from $T(n,r)$ in $o(n^2)$ edges, uniformly all $\lceil(r-1)n/r\rceil\le\Delta\le n-1$. We further establish a general spectral transfer principle: for finite forbidden families, a decomposition-family bound of order $O(n^{1+s})$ yields a spectral bound with error $O(n^s)$ for $0\le s<1$.