An integer linear system (ILS) is a linear system with integer constraints. The solution graph of an ILS is defined as an undirected graph defined on the set of feasible solutions to the ILS. A pair of feasible solutions is connected by an edge in the solution graph if the Hamming distance between them is 1. We consider a property of the coefficient matrix of an ILS such that the solution graph is connected for any right-hand side vector. Especially, we focus on the existence of an elimination ordering (EO) of a coefficient matrix, which is known as the sufficient condition for the connectedness of the solution graph for any right-hand side vector. That is, we consider the question whether the existence of an EO of the coefficient matrix is a necessary condition for the connectedness of the solution graph for any right-hand side vector. We first prove that if a coefficient matrix has at least four rows and at least three columns, then the existence of an EO may not be a necessary condition. Next, we prove that if a coefficient matrix has at most three rows or at most two columns, then the existence of an EO is a necessary condition.

翻译：整数线性系统（ILS）是一种带有整数约束的线性系统。其解图被定义为一个无向图，其顶点集为ILS的所有可行解。若两个可行解之间的汉明距离为1，则它们在解图中由一条边相连。我们研究ILS系数矩阵的一种性质，该性质保证对于任意右端向量，其解图都是连通的。特别地，我们关注系数矩阵消元序（EO）的存在性，已知这是解图对任意右端向量保持连通的充分条件。即，我们探讨系数矩阵EO的存在性是否也是解图对任意右端向量保持连通的必要条件。我们首先证明，若系数矩阵至少有四行且至少有三列，则EO的存在性可能不是必要条件。其次，我们证明若系数矩阵至多有三行或至多有两列，则EO的存在性是必要条件。