Uniquely Restricted Matchings and Edge Colorings
نویسندگان
چکیده
A matching in a graph is uniquely restricted if no other matching covers exactly the same set of vertices. This notion was defined by Golumbic, Hirst, and Lewenstein and studied in a number of articles. Our contribution is twofold. We provide approximation algorithms for computing a uniquely restricted matching of maximum size in some bipartite graphs. In particular, we achieve a ratio of 9/5 for subcubic bipartite graphs, improving over a 2-approximation algorithm proposed by Mishra. Furthermore, we study the uniquely restricted chromatic index of a graph, defined as the minimum number of uniquely restricted matchings into which its edge set can be partitioned. We provide tight upper bounds in terms of the maximum degree and characterize all extremal graphs. Our constructive proofs yield efficient algorithms to determine the corresponding edge colorings.
منابع مشابه
Perfect Matchings in Edge-Transitive Graphs
We find recursive formulae for the number of perfect matchings in a graph G by splitting G into subgraphs H and Q. We use these formulas to count perfect matching of P hypercube Qn. We also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is , where denotes the number of perfect matchings in G, is the graph constructed from by deleting edges with an en...
متن کاملUniquely Restricted Matchings in Interval Graphs
A matching M in a graph G is said to be uniquely restricted if there is no other matching in G that matches the same set of vertices as M . We describe a polynomial-time algorithm to compute a maximum cardinality uniquely restricted matching in an interval graph, thereby answering a question of Golumbic et al. (“Uniquely restricted matchings”, M. C. Golumbic, T. Hirst and M. Lewenstein, Algorit...
متن کاملTriangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids
A matchingM is uniquely restricted in a graph G if its saturated vertices induce a subgraph which has a unique perfect matching, namely M itself [M.C. Golumbic, T. Hirst, M. Lewenstein, Uniquely restricted matchings, Algorithmica 31 (2001) 139–154]. G is a König–Egerváry graph provided (G)+ (G)= |V (G)| [R.W. Deming, Independence numbers of graphs—an extension of the König–Egerváry theorem, Dis...
متن کاملAnti-Ramsey Problems for t Edge-Disjoint Rainbow Spanning Subgraphs: Cycles, Matchings, or Trees
We seek the maximum number of colors in an edge-coloring of the complete graph Kn not having t edge-disjoint rainbow spanning subgraphs of specified types. Let c(n, t), m(n, t), and r(n, t) denote the answers when the spanning subgraphs are cycles, matchings, or trees, respectively. We prove c(n, t) = ( 2 ) + t for n ≥ 8t − 1 and m(n, t) = ( 2 ) + t for n > 4t + 10. We prove r(n, t) = ( 2 ) + t...
متن کاملThe strong edge colorings of a sparse random graph
The strong chromatic index of a graph G is the smallest integer k such that the edge set E( G) can be partitioned into k induced subgraphs of G which form matchings. In this paper we consider the behavior of the strong chromatic index of a sparse random graph K (n, p), where p = p(n) = 0(1).
متن کامل