Edge-choosability of multicircuits

AbstractA multicircuit is a multigraph whose underlying simple graph is a circuit (a connected 2-regular graph). The List-Colouring Conjecture (LCC) is that every multigraph G has edge-choosability (list chromatic index) ch′(G) equal to its chromatic index χ′(G). In this paper the LCC is proved first for multicircuits, and then, building on results of Peterson and Woodall, for any multigraph G in which every block is bipartite or a multicircuit or has at most four vertices or has underlying simple graph of the form K1, 1, p

Preference Swaps for the Stable Matching Problem

An instance $I$ of the Stable Matching Problem (SMP) is given by a bipartite graph with a preference list of neighbors for every vertex. A swap in $I$ is the exchange of two consecutive vertices in a preference list. A swap can be viewed as a smallest perturbation of $I$. Boehmer et al. (2021) designed a polynomial-time algorithm to find the minimum number of swaps required to turn a given maximal matching into a stable matching. To generalize this result to the many-to-many version of SMP, we introduce a new representation of SMP as an extended bipartite graph and reduce the problem to submodular minimization. It is a natural problem to establish computational complexity of deciding whether at most $k$ swaps are enough to turn $I$ into an instance where one of the maximum matchings is stable. Using a hardness result of Gupta et al. (2020), we prove that it is NP-hard to decide whether at most $k$ swaps are enough to turn $I$ into an instance with a stable perfect matching. Moreover, this problem parameterized by $k$ is W[1]-hard. We also obtain a lower bound on the running time for solving the problem using the Exponential Time Hypothesis