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Odd decompositions and coverings of graphs. (English) Zbl 1458.05217
Summary: A (finite) graph is odd if all its vertices have odd degrees. The principal aim of this survey is to present the current state of research on covers and decompositions of graphs into fewest possible number of odd subgraphs. Given a graph \(G\), the parameters \(\chi_o^\prime(G)\) and \(\operatorname{cov}_o(G)\) denote, respectively, the minimum size of a decomposition and cover of \(G\) consisting of odd subgraphs. L. Pyber [in: Sets, graphs and numbers. A birthday salute to Vera T. Sós and András Hajnal. Amsterdam: North-Holland Publishing Company. 583–610 (1992; Zbl 0792.05110)] and T. Mátrai [J. Graph Theory 53, No. 1, 77–82 (2006; Zbl 1098.05067)], respectively, have shown that for every simple graph \(G\) it holds that \(\chi_o^\prime(G)\leq 4\) and \(\operatorname{cov}_o(G)\leq 3\), with both bounds being sharp. The multigraph analogues of the same inequalities, given by the present authors in [ J. Graph Theory 92, No. 3, 304–321 (2019; Zbl 1429.05169); Adv. Math., Sci. J. 8, No. 2, 63–68 (2019; Zbl 1431.05124)], respectively, are discussed in detail. The list versions of the graph parameters \(\chi_o^\prime(G)\) and \(\operatorname{cov}_o (G)\), along with other generalizations and possible new directions of related research, are considered in the latter part of the article. Throughout we also pose various structural and algorithmic questions and problems.
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
Full Text: DOI
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