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 Bul. Acad. Ştiinţe Repub. Mold. Mat., 2012, Number 1, Pages 50–58 (Mi basm303)  On cyclically-interval edge colorings of trees

R. R. Kamalian

Institute for Informatics and Automation Problems, National Academy of Sciences of RA, Yerevan, Republic of Armenia

Abstract: For an undirected, simple, finite, connected graph $G$, we denote by $V(G)$ and $E(G)$ the sets of its vertices and edges, respectively. A function $\varphi\colon E(G)\to\{1,2,…,t\}$ is called a proper edge $t$-coloring of a graph $G$ if adjacent edges are colored differently and each of $t$ colors is used. An arbitrary nonempty subset of consecutive integers is called an interval. If $\varphi$ is a proper edge $t$-coloring of a graph $G$ and $x\in V(G)$, then $S_G(x,\varphi)$ denotes the set of colors of edges of $G$ which are incident with $x$. A proper edge $t$-coloring $\varphi$ of a graph $G$ is called a cyclically-interval $t$-coloring if for any $x\in V(G)$ at least one of the following two conditions holds: a) $S_G(x,\varphi)$ is an interval, b) $\{1,2,…,t\}\setminus S_G(x,\varphi)$ is an interval. For any $t\in\mathbb N$, let $\mathfrak M_t$ be the set of graphs for which there exists a cyclically-interval $t$-coloring, and let $\mathfrak M\equiv\bigcup_{t\geq1}\mathfrak M_t$. For an arbitrary tree $G$, it is proved that $G\in\mathfrak M$ and all possible values of $t$ are found for which $G\in\mathfrak M_t$.

Keywords and phrases: tree, interval edge coloring, cyclically-interval edge coloring. Full text: PDF file (137 kB) References: PDF file   HTML file

Bibliographic databases:  MSC: 05C05, 05C15
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Citation: R. R. Kamalian, “On cyclically-interval edge colorings of trees”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2012, no. 1, 50–58 Citation in format AMSBIB
\Bibitem{Kam12}
\by R.~R.~Kamalian
\paper On cyclically-interval edge colorings of trees
\jour Bul. Acad. \c Stiin\c te Repub. Mold. Mat.
\yr 2012
\issue 1
\pages 50--58
\mathnet{http://mi.mathnet.ru/basm303}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2987326}
\zmath{https://zbmath.org/?q=an:1252.05066}

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This publication is cited in the following articles:
1. R. Kamalian, “On a Number of Colors in Cyclically Interval Edge Colorings of Simple Cycles”, Open Journal of Discrete Mathematics, 3:1 (2013), 43–48  2. Bodur M., Luedtke J.R., “Integer Programming Formulations For Minimum Deficiency Interval Coloring”, Networks, 72:2 (2018), 249–271     • Number of views: This page: 137 Full text: 30 References: 29 First page: 1 Contact us: math-net2019_10 [at] mi-ras ru Terms of Use Registration Logotypes © Steklov Mathematical Institute RAS, 2019