Breuer and Klivans defined a diverse class of scheduling problems in terms of Boolean formulas with atomic clauses that are inequalities. We consider what we call graph-like scheduling problems. These are Boolean formulas that are conjunctions of disjunctions of atomic clauses $(x_i \neq x_j)$. These problems generalize proper coloring in graphs and hypergraphs. We focus on the existence of a solution with all $x_i$ taking the value of $0$ or $1$ (i.e. problems analogous to the bipartite case). When a graph-like scheduling problem has such a solution, we say it has property B just as is done for $2$-colorable hypergraphs. We define the notion of a $\lambda$-uniform graph-like scheduling problem for any integer partition $\lambda$. Some bounds are attained for the size of the smallest $\lambda$-uniform graph-like scheduling problems without property B. We make use of both random and constructive methods to obtain bounds. Just as in the case of hypergraphs finding tight bounds remains an open problem.

翻译：Breuer和Klivans通过包含不等式原子子句的布尔公式定义了一类多样的调度问题。我们考虑所谓类图调度问题，即由原子子句$(x_i \neq x_j)$的析取式构成的合取布尔公式。这类问题推广了图与超图中的正常着色概念。我们关注所有$x_i$取值为$0$或$1$的解的存在性（即类似于二部图情形的问题）。当类图调度问题存在此类解时，我们称其具有性质B，正如$2$-可着色超图的术语定义。对于任意整数划分$\lambda$，我们定义了$\lambda$-一致类图调度问题的概念。对于无性质B的最小$\lambda$-一致类图调度问题的大小，我们得到了若干界。通过随机方法与构造性方法相结合获得这些界。与超图情形类似，寻求紧界仍是未解决之问题。