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 nn-dimensional cube

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 $\chi'(G)\leq t\leq |E(G)|$, 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 $G$: $$\alpha(G)\equiv\bigcup_{t=\chi'(G)}^{|E(G)|}\alpha(G,t).$$ An arbitrary nonempty finite subset of consecutive integers is called an interval. If $\varphi\in\alpha(G)$ and $x\in V(G)$, 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 $\varphi\in\alpha(G)$, then define $f_G(\varphi)\equiv|\{x\in V(G)/S_G(x,\varphi) \textrm{is an interval}\}|$. For a graph $G$ and any integer $t$, satisfying the inequality $\chi'(G)\leq t\leq |E(G)|$, we define: $$\mu_1(G,t)\equiv\min_{\varphi\in\alpha(G,t)}f_G(\varphi),\qquad \mu_2(G,t)\equiv\max_{\varphi\in\alpha(G,t)}f_G(\varphi).$$ For any graph $G$, we set: $$\mu_{11}(G)\equiv\min_{\chi'(G)\leq t\leq|E(G)|}\mu_1(G,t),\qquad \mu_{12}(G)\equiv\max_{\chi'(G)\leq t\leq|E(G)|}\mu_1(G,t),$$ $$\mu_{21}(G)\equiv\min_{\chi'(G)\leq t\leq|E(G)|}\mu_2(G,t),\qquad \mu_{22}(G)\equiv\max_{\chi'(G)\leq t\leq|E(G)|}\mu_2(G,t).$$ 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.Comment: 9 pages. arXiv admin note: substantial text overlap with arXiv:1205.012

Some remarks on relations between the μ\mu-parameters of regular graphs

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 $\chi'(G)\leq t\leq |E(G)|$, 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 $G$: $$\alpha(G)\equiv\bigcup_{t=\chi'(G)}^{|E(G)|}\alpha(G,t).$$ An arbitrary nonempty finite subset of consecutive integers is called an interval. If $\varphi\in\alpha(G)$ and $x\in V(G)$, 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 $\varphi\in\alpha(G)$, then define $f_G(\varphi)\equiv|\{x\in V(G)/S_G(x,\varphi) \textrm{is an interval}\}|$. For a graph $G$ and any integer $t$, satisfying the inequality $\chi'(G)\leq t\leq |E(G)|$, we define: $$\mu_1(G,t)\equiv\min_{\varphi\in\alpha(G,t)}f_G(\varphi),\qquad \mu_2(G,t)\equiv\max_{\varphi\in\alpha(G,t)}f_G(\varphi).$$ For any graph $G$, we set: $$\mu_{11}(G)\equiv\min_{\chi'(G)\leq t\leq|E(G)|}\mu_1(G,t),\qquad \mu_{12}(G)\equiv\max_{\chi'(G)\leq t\leq|E(G)|}\mu_1(G,t),$$ $$\mu_{21}(G)\equiv\min_{\chi'(G)\leq t\leq|E(G)|}\mu_2(G,t),\qquad \mu_{22}(G)\equiv\max_{\chi'(G)\leq t\leq|E(G)|}\mu_2(G,t).$$ For regular graphs, some relations between the $\mu$-parameters are obtained.Comment: arXiv admin note: text overlap with arXiv:1307.1389, arXiv:1307.234

On the μ\mu-parameters of the Petersen graph

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 $\chi'(G)\leq t\leq |E(G)|$, 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 $G$: $$\alpha(G)\equiv\bigcup_{t=\chi'(G)}^{|E(G)|}\alpha(G,t).$$ An arbitrary nonempty finite subset of consecutive integers is called an interval. If $\varphi\in\alpha(G)$ and $x\in V(G)$, 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 $\varphi\in\alpha(G)$, then define $f_G(\varphi)\equiv|\{x\in V(G)/S_G(x,\varphi) \textrm{is an interval}\}|$. For a graph $G$ and any integer $t$, satisfying the inequality $\chi'(G)\leq t\leq |E(G)|$, we define: $$\mu_1(G,t)\equiv\min_{\varphi\in\alpha(G,t)}f_G(\varphi),\qquad \mu_2(G,t)\equiv\max_{\varphi\in\alpha(G,t)}f_G(\varphi).$$ For any graph $G$, we set: $$\mu_{11}(G)\equiv\min_{\chi'(G)\leq t\leq|E(G)|}\mu_1(G,t),\qquad \mu_{12}(G)\equiv\max_{\chi'(G)\leq t\leq|E(G)|}\mu_1(G,t),$$ $$\mu_{21}(G)\equiv\min_{\chi'(G)\leq t\leq|E(G)|}\mu_2(G,t),\qquad \mu_{22}(G)\equiv\max_{\chi'(G)\leq t\leq|E(G)|}\mu_2(G,t).$$ For the Petersen graph, the exact values of the parameters $\mu_{11}$, $\mu_{12}$, $\mu_{21}$ and $\mu_{22}$ are found.Comment: arXiv admin note: substantial text overlap with arXiv:1307.1389; and substantial text overlap with arXiv:1205.0125, arXiv:1307.1389 by other author