arXiv subjects

N. N. Davtyan

Publications and source records attributed to N. N. Davtyan.

3 recordsLinked to original sources

arXiv2013

Some remarks on relations between the $μ$-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 $φ:E(G)\rightarrow \{1,...,t\}$ 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 $χ'(G)$. For any graph $G$, and for any integer $t$ satisfying the inequality $χ'(G)\leq t\leq |E(G)|$, we denote by $α(G,t)$ the set of all proper edge $t$-colorings of $G$. Let us also define a set $α(G)$ of all proper edge colorings of a graph $G$: $$ α(G)\equiv\bigcup_{t=χ'(G)}^{|E(G)|}α(G,t). $$ An arbitrary nonempty finite subset of consecutive integers is called an interval. If $φ\inα(G)$ and $x\in V(G)$, then the set of colors of edges of $G$ which are incident with $x$ is denoted by $S_G(x,φ)$ and is called a spectrum of the vertex $x$ of the graph $G$ at the proper edge coloring $φ$. If $G$ is a graph and $φ\inα(G)$, then define $f_G(φ)\equiv|\{x\in V(G)/S_G(x,φ) \textrm{is an interval}\}|$. For a graph $G$ and any integer $t$, satisfying the inequality $χ'(G)\leq t\leq |E(G)|$, we define: $$ μ_1(G,t)\equiv\min_{φ\inα(G,t)}f_G(φ),\qquad μ_2(G,t)\equiv\max_{φ\inα(G,t)}f_G(φ). $$ For any graph $G$, we set: $$ μ_{11}(G)\equiv\min_{χ'(G)\leq t\leq|E(G)|}μ_1(G,t),\qquad μ_{12}(G)\equiv\max_{χ'(G)\leq t\leq|E(G)|}μ_1(G,t), $$ $$ μ_{21}(G)\equiv\min_{χ'(G)\leq t\leq|E(G)|}μ_2(G,t),\qquad μ_{22}(G)\equiv\max_{χ'(G)\leq t\leq|E(G)|}μ_2(G,t). $$ For regular graphs, some relations between the $μ$-parameters are obtained.

cs.DM↗

arXiv2013

On the $μ$-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 $φ:E(G)\rightarrow \{1,...,t\}$ 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 $χ'(G)$. For any graph $G$, and for any integer $t$ satisfying the inequality $χ'(G)\leq t\leq |E(G)|$, we denote by $α(G,t)$ the set of all proper edge $t$-colorings of $G$. Let us also define a set $α(G)$ of all proper edge colorings of a graph $G$: $$ α(G)\equiv\bigcup_{t=χ'(G)}^{|E(G)|}α(G,t). $$ An arbitrary nonempty finite subset of consecutive integers is called an interval. If $φ\inα(G)$ and $x\in V(G)$, then the set of colors of edges of $G$ which are incident with $x$ is denoted by $S_G(x,φ)$ and is called a spectrum of the vertex $x$ of the graph $G$ at the proper edge coloring $φ$. If $G$ is a graph and $φ\inα(G)$, then define $f_G(φ)\equiv|\{x\in V(G)/S_G(x,φ) \textrm{is an interval}\}|$. For a graph $G$ and any integer $t$, satisfying the inequality $χ'(G)\leq t\leq |E(G)|$, we define: $$ μ_1(G,t)\equiv\min_{φ\inα(G,t)}f_G(φ),\qquad μ_2(G,t)\equiv\max_{φ\inα(G,t)}f_G(φ). $$ For any graph $G$, we set: $$ μ_{11}(G)\equiv\min_{χ'(G)\leq t\leq|E(G)|}μ_1(G,t),\qquad μ_{12}(G)\equiv\max_{χ'(G)\leq t\leq|E(G)|}μ_1(G,t), $$ $$ μ_{21}(G)\equiv\min_{χ'(G)\leq t\leq|E(G)|}μ_2(G,t),\qquad μ_{22}(G)\equiv\max_{χ'(G)\leq t\leq|E(G)|}μ_2(G,t). $$ For the Petersen graph, the exact values of the parameters $μ_{11}$, $μ_{12}$, $μ_{21}$ and $μ_{22}$ are found.

cs.DM↗

arXiv2013

An inequality for the number of vertices with an interval spectrum in edge labelings of regular graphs

We consider undirected simple finite graphs. The sets of vertices and edges of a graph $G$ are denoted by $V(G)$ and $E(G)$, respectively. For a graph $G$, we denote by $δ(G)$ and $η(G)$ the least degree of a vertex of $G$ and the number of connected components of $G$, respectively. For a graph $G$ and an arbitrary subset $V_0\subseteq V(G)$ $G[V_0]$ denotes the subgraph of the graph $G$ induced by the subset $V_0$ of its vertices. An arbitrary nonempty finite subset of consecutive integers is called an interval. A function $φ:E(G)\rightarrow \{1,2,\dots,|E(G)|\}$ is called an edge labeling of the graph $G$, if for arbitrary different edges $e'\in E(G)$ and $e''\in E(G)$, the inequality $φ(e')\neq φ(e'')$ holds. If $G$ is a graph, $x$ is its arbitrary vertex, and $φ$ is its arbitrary edge labeling, then the set $S_G(x,φ)\equiv\{φ(e)/ e\in E(G), e \textrm{is incident with} x$\} is called a spectrum of the vertex $x$ of the graph $G$ at its edge labeling $φ$. If $G$ is a graph and $φ$ is its arbitrary edge labeling, then $V_{int}(G,φ)\equiv\{x\in V(G)/\;S_G(x,φ)\textrm{is an interval}\}$. For an arbitrary $r$-regular graph $G$ with $r\geq2$ and its arbitrary edge labeling $φ$, the inequality $$ |V_{int}(G,φ)|\leq\bigg\lfloor\frac{3\cdot|V(G)|-2\cdotη(G[V_{int}(G,φ)])}{4}\bigg\rfloor. $$ is proved.

math.CO↗