Geodesic
On a metric property of perfect colorings
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 640-646

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Given a perfect coloring of a graph, we prove that the $L_1$ distance between two rows of the adjacency matrix of the graph is not less than the $L_1$ distance between the corresponding rows of the parameter matrix of the coloring. With the help of an algebraic approach, we deduce corollaries of this result for perfect $2$-colorings and perfect colorings in distance-$l$ graphs and distance-regular graphs. We also provide examples of infinite graphs, where the obtained property rejects several putative parameter matrices of perfect colorings.
Keywords: perfect coloring, perfect structure, square grid, triangular grid.
Mots-clés : $L_1$ distance, circulant graph 
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     author = {A. A. Taranenko},
     title = {On a metric property of perfect colorings},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {640--646},
     publisher = {mathdoc},
     volume = {18},
     number = {1},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2021_18_1_a20/}
}
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A. A. Taranenko. On a metric property of perfect colorings. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 640-646. http://geodesic.mathdoc.fr/item/SEMR_2021_18_1_a20/