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<p>A&#xa0;chain is a&#xa0;tree where two vertices have degree&#xa0;1 and all others have degree&#xa0;2. A&#xa0;propeller is a&#xa0;tree that has one vertex of degree&#xa0;3, three vertices of degree&#xa0;1, and all other vertices have degree&#xa0;2. A&#xa0;proper propeller is a&#xa0;propeller, where vertices of degree one are at equal distances from the vertex of degree&#xa0;3. We study the following problem: how to find the number of 3-colorings of a&#xa0;chain and a&#xa0;proper propeller in the case where the numbers of vertices of each color are given? In both cases, generating functions are presented.</p>

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ON 3-COLORING OF CHAINS AND PROPELLERS

  • 期刊论文

  • 发表日期：

    2026年1月1日

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  • Yu. Yu. Kochetkov

摘要

A chain is a tree where two vertices have degree 1 and all others have degree 2. A propeller is a tree that has one vertex of degree 3, three vertices of degree 1, and all other vertices have degree 2. A proper propeller is a propeller, where vertices of degree one are at equal distances from the vertex of degree 3. We study the following problem: how to find the number of 3-colorings of a chain and a proper propeller in the case where the numbers of vertices of each color are given? In both cases, generating functions are presented.