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Partitions of Multigraphs Without \(C_4\)

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Abstract

Let G be a graph that may have multiedges, let \(\mu (u, v)\) denote the number of edges joining u and v in G, and let \(\mu (u)=\max _{v\in N(u)} \mu (u, v)\). Let \(a, b:V(G)\ \longrightarrow {\mathbb {N}}{\setminus }\{0,1\}\) be two functions with \(d_G(v)\ge a(v)+b(v)+2\mu (v)-3\) for each \(v\in V(G)\). We show that, if G does not contain \(C_4\), then G admits a partition (AB) with \(d_{G[A]}(u)\ge a(u)\) for each \(u\in A\), and \(d_{G[B]}(v)\ge b(v)\) for each \(v\in B\). This generalizes a theorem of Ma and Yang on simple graphs, and answers a question of Schweser and Stiebitz.

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Correspondence to Baogang Xu.

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Partially supported by NSFC projects 11931006.

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Song, J., Xu, B. Partitions of Multigraphs Without \(C_4\). Graphs and Combinatorics 37, 2095–2111 (2021). https://doi.org/10.1007/s00373-021-02335-4

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