@article{oai:ynu.repo.nii.ac.jp:00011354,
 author = {Enami, Kengo and Ozeki, Kenta and Yamaguchi, Tomoki},
 journal = {Graphs and combinatorics},
 month = {Jun},
 note = {We consider a proper coloring of a plane graph such that no face is rainbow, where a face is rainbow if any two vertices on its boundary have distinct colors. Such a coloring is said to be proper anti-rainbow. A plane quadrangulation G is a plane graph in which all faces are bounded by a cycle of length 4. In this paper, we show that the number of colors in a proper anti-rainbow coloring of a plane quadrangulation G does not exceed 3α(G)/2, where α(G) is the independence number of G. Moreover, if the minimum degree of G is 3 or if G is 3-connected, then this bound can be improved to 5α(G)/4 or 7α(G)/6+1/3, respectively. All of these bounds are tight., This is a post-peer-review, pre-copyedit version of an article published in Graphs and Combinatorics. The final authenticated version is available online at: http://dx.doi.org/10.1007/s00373-021-02350-5},
 title = {Proper Colorings of Plane Quadrangulations Without Rainbow Faces},
 year = {2021}
}