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- W1635226503 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 $varphi:E(G)rightarrow {1,...,t}$ is called a proper edge $t$-coloring of a graph $G$, if adjacent edges are colored differently and each of $t$ colors is used. The least value of $t$ for which there exists a proper edge $t$-coloring of a graph $G$ is denoted by $chi'(G)$. For any graph $G$, and for any integer $t$ satisfying the inequality $chi'(G)leq tleq |E(G)|$, we denote by $alpha(G,t)$ the set of all proper edge $t$-colorings of $G$. Let us also define a set $alpha(G)$ of all proper edge colorings of a graph $G$: $$ alpha(G)equivbigcup_{t=chi'(G)}^{|E(G)|}alpha(G,t). $$ An arbitrary nonempty finite subset of consecutive integers is called an interval. If $varphiinalpha(G)$ and $xin V(G)$, then the set of colors of edges of $G$ which are incident with $x$ is denoted by $S_G(x,varphi)$ and is called a spectrum of the vertex $x$ of the graph $G$ at the proper edge coloring $varphi$. If $G$ is a graph and $varphiinalpha(G)$, then define $f_G(varphi)equiv|{xin V(G)/S_G(x,varphi) textrm{is an interval}}|$. For a graph $G$ and any integer $t$, satisfying the inequality $chi'(G)leq tleq |E(G)|$, we define: $$ mu_1(G,t)equivmin_{varphiinalpha(G,t)}f_G(varphi),qquad mu_2(G,t)equivmax_{varphiinalpha(G,t)}f_G(varphi). $$ For any graph $G$, we set: $$ mu_{11}(G)equivmin_{chi'(G)leq tleq|E(G)|}mu_1(G,t),qquad mu_{12}(G)equivmax_{chi'(G)leq tleq|E(G)|}mu_1(G,t), $$ $$ mu_{21}(G)equivmin_{chi'(G)leq tleq|E(G)|}mu_2(G,t),qquad mu_{22}(G)equivmax_{chi'(G)leq tleq|E(G)|}mu_2(G,t). $$ For any positive integer $n$, the exact values of the parameters $mu_{11}$, $mu_{12}$, $mu_{21}$ and $mu_{22}$ are found for the graph of the $n$-dimensional cube." @default.
- W1635226503 created "2016-06-24" @default.
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- W1635226503 date "2013-07-04" @default.
- W1635226503 modified "2023-09-26" @default.
- W1635226503 title "On the extremal values of the number of vertices with an interval spectrum on the set of proper edge colorings of the graph of the $n$-dimensional cube" @default.
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- W1635226503 doi "https://doi.org/10.48550/arxiv.1307.1389" @default.
- W1635226503 hasPublicationYear "2013" @default.
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