Extremal problems on $[a, b]$-covered graphs

A graph $G$ is $[a,b]$-covered if for each edge $e$ of $G$ there is an $[a,b]$-factor containing it. For $a=b=1$, an $[a,b]$-covered graph is a matching covered graph. The structural theory of matching covered graphs constitutes a cornerstone of modern matching theory. Determining whether a given graph is matching covered is a fundamental problem in structural graph theory. Lucchesi et al. [SIAM J. Discrete Math., 2018] showed that a connected graph $G$ is matching covered if and only if every barrier of $G$ is a stable set. In this paper, we completely characterize the extremal graphs that maximize the size or the spectral radius among all non-matching-covered graphs. For $a \leq b$ and $b \geq 2,$ Hao and Li [Electron. J. Combin., 2024] investigated the extremal problems on $[a,b]$-factor graphs: If $G$ contains no $[a,b]$-factors, then $e(G)\leq \binom{n-1}{2}+a-1$ with equality if and only if $G\cong H_{n,a},$ where $H_{n,a} = K_{a-1} \vee (K_{n-a} \cup K_1).$ Moreover, if $G$ contains no $[a,b]$-factors, then $\rho(G)\leq \rho(H_{n,a})$ with equality if and only if $G \cong H_{n,a}.$ Judging from the structral characterization, non-$[a,b]$-covered graphs exhibit highly complex structures, making the associated extremal problems significantly challenging. To overcome this, we develop a novel minimum-degree forcing technique. Combining this technique and spectral-structural analysis, we in this paper provide complete characterizations of the extremal graphs that maximize the size or the spectral radius within the set of non-$[a,b]$-covered graphs. An intriguing phenomenon revealed by our results is that $H_{n,a}$ remains both the size-extremal graph and the spectral extremal graph for this larger set of non-$[a,b]$-covered graphs. Consequently, our results strengthen the results of Hao-Li.