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Maximum properly colored trees in edge-colored graphs

Abstract

An edge-colored graph G is a graph with an edge coloring. We say G is properly colored if any two adjacent edges of G have distinct colors, and G is rainbow if any two edges of G have distinct colors. For a vertex \(v \in V(G)\), the color degree \(d_G^{col}(v)\) of v is the number of distinct colors appearing on edges incident with v. The minimum color degree \(\delta ^{col}(G)\) of G is the minimum \(d_G^{col}(v)\) over all vertices \(v \in V(G)\). In this paper, we study the relation between the order of maximum properly colored tree in G and the minimum color degree \(\delta ^{col}(G)\) of G. We obtain that for an edge-colored connected graph G, the order of maximum properly colored tree is at least \(\min \{|G|, 2\delta ^{col}(G)\}\), which generalizes the result of Cheng et al. [Properly colored spanning trees in edge-colored graphs, Discrete Math., 343 (1), 2020]. Moreover, the lower bound \(2\delta ^{col}(G)\) in our result is sharp and we characterize all extremal graphs G with the maximum properly colored tree of order \(2\delta ^{col}(G) \ne |G|\).

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Acknowledgements

The first author is supported by the China Scholarship Council (No. 201806220051). The third author is supported by Grant for Overseas Challenge Program for Young Researchers from JSPS (No. 201980222) and JSPS KAKENHI (No. JP20J15332).

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Correspondence to Shun-ichi Maezawa.

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Hu, J., Li, H. & Maezawa, Si. Maximum properly colored trees in edge-colored graphs. J Comb Optim (2021). https://doi.org/10.1007/s10878-021-00824-z

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Keywords

  • Edge-colored graph
  • Properly colored tree
  • Color degree