a b s t r a c tAn induced matching in a graph G = (V , E) is a matching M such that (V , M) is an induced subgraph of G. Clearly, among two vertices with the same neighbourhood (called twins) at most one is matched in any induced matching, and if one of them is matched then there is another matching of the same size that matches the other vertex. Motivated by this, Kanj et al.[10] studied induced matchings in twinless graphs. They showed that any twinless planar graph contains an induced matching of size at least n 40 and that there are twinless planar graphs that do not contain an induced matching of size greater than n 27 + O(1). We improve both these bounds to n 28 + O(1), which is tight up to an additive constant. This implies that the problem of deciding whether a planar graph has an induced matching of size k has a kernel of size at most 28k. We also show for the first time that this problem is fixed parameter tractable for graphs of bounded arboricity. Kanj et al. also presented an algorithm which decides in O(2 159 √ k + n)-time whether an n-vertex planar graph contains an induced matching of size k. Our results improve the time complexity analysis of their algorithm. However, we also show a more efficient O(2 25.5 √ k + n)-time algorithm. Its main ingredient is a new, O * (4 l )-time algorithm for finding a maximum induced matching in a graph of branch width at most l.
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It is conjectured that every fullerene graph is hamiltonian. Jendrol' and Owens proved [J. Math. Chem. 18 (1995), pp. 83-90] that every fullerene graph on n vertices has a cycle of length at least 4n/5. Recently, Král' et al. improved it to 5n/6 − 2/3. In this paper, we study 2-factors of fullerene graphs. As a by-product, we get an improvement of a lower bound on the length of the longest cycle in a fullerene graph. We present a constructive proof of the bound 6n/7 + 2/7. * Supported in part by bilateral project BI-SK/05-07-001 between Slovenia and Slovakia. † Supported in part by bilateral project SK-SI-00806 between Slovakia and Slovenia.
Let G be a plane graph with maximum face size ∆ * . If all faces of G with size four or more are vertex disjoint, then G has a cyclic coloring with ∆ * + 1 colors, i.e., a coloring such that all vertices incident with the same face receive distinct colors. * This research was supported by the Czech-Slovenian bilateral project MEB 090805 (on the Czech side) and BI-CZ/08-09-005 (on the Slovenian side).
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