The main problem in the area of graph property testing is to understand which graph properties are \emph{testable}, which means that with constantly many queries to any input graph $G$, a tester can decide with good probability whether $G$ satisfies the property, or is far from satisfying the property. Testable properties are well understood in the dense model and in the bounded degree model, but little is known in sparse graph classes when graphs are allowed to have unbounded degree. This is the setting of the \emph{sparse model}. We prove that for any proper minor-closed class $\mathcal{G}$, any monotone property (i.e., any property that is closed under taking subgraphs) is testable for graphs from $\mathcal{G}$ in the sparse model. This extends a result of Czumaj and Sohler (FOCS'19), who proved it for monotone properties with finitely many forbidden subgraphs. Our result implies for instance that for any integers $k$ and $t$, $k$-colorability of $K_t$-minor free graphs is testable in the sparse model. Elek recently proved that monotone properties of bounded degree graphs from minor-closed classes that are closed under disjoint union can be verified by an approximate proof labeling scheme in constant time. We show again that the assumption of bounded degree can be omitted in his result.

翻译：图形属性测试领域的主要问题是了解哪些图形属性是 \ emph{ 测试}, 这意味着,如果对任何输入图形进行多次查询, 测试者可以非常概率地决定$G$是否满足属性, 或者远不能满足属性。 在密度模型和约束度模型中, 测试属性在密度模型和约束度模型中是完全理解的, 但是在允许图形不受限制度时, 稀薄的图形类别中却鲜为人知。 这是设置 \ emph{ 偏差模型 。 这就是 。 我们证明, 对于任何合适的小封闭类, $\ mathcal{ G} 美元, 测试者可以确定任何单一属性( 即, 在子图下关闭的属性是否满足, 或远不满足 。 在 $_ group- col- deal 模型中, 能够再次证明单项的单项属性属性。 我们的任意整数 $美元和 $美元- 美元- colorg- be contradeal condeal deal deal deal deal deal rogal deal ex ex ex ex exmal exmal roal romal roup ex ex ex rol rol ex ex ex ex ex ex ex ex ex ex roup ex ex ex ex ex ex ex rol rol rol ex rol rol rol rol rol rol rol rol rol rol rol rol rol rol rol ex, ex exgyl exl exl ex exl ex exl exl exl exl ex ex ex ex ex ex ex ex ex ex ex ex exl ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex