Equitable vertex arboricity of d-degenerate graphs
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A minimization problem in graph theory so-called the equitable tree-coloring problem can be used to formulate a structure decomposition problem on the communication network with some security considerations. Precisely, an equitable tree-k-coloring of a graph is a vertex coloring using k distinct colors such that every color class induces a forest and the sizes of any two color classes differ by at most one. In this paper, we establish some theoretical results on the equitable tree-colorings of graphs by showing that every d-degenerate graph with maximum degree at most Δ is equitably tree-k-colorable for every integer k≥ (Δ+1)/2 provided that Δ≥ 10d. This generalises the result of Chen et al.[J. Comb. Optim. 34(2) (2017) 426–432] which states that every 5-degenerate graph with maximum degree at most Δ is equitably tree-k-colorable for every integer k≥ (Δ+1)/2.
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