Combinatorial auctions (CAs) offer the flexibility for bidders to articulate complex preferences when competing for multiple assets. However, the behavior of bidders under different payment rules is often unclear. Our research explores the relationship between core constraints and several core-selecting payment rules. Specifically, we examine the natural and desirable property of payment rules of being non-decreasing, which ensures that bidding higher does not lead to lower payments. Earlier studies revealed that the VCG-nearest payment method – a commonly employed payment rule – fails to adhere to this principle even for single-minded CAs. We establish that when a single effective core constraint exists, the payment maintains the non-decreasing property in single-minded CAs. To identify auctions where such a constraint is present, we introduce a novel framework using conflict graphs to represent single-minded CAs and establish sufficient conditions for the existence of single effective core constraints. We proceed with an analysis of the implications on bidder behavior, demonstrating that there is no overbidding in any Nash equilibrium when considering non-decreasing core-selecting payment rules. Our study concludes by establishing the non-decreasing nature of two additional payment rules, namely the proxy and proportional payment rules, for single-minded CAs.