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In this paper, we consider the linear and circular consecutive $k$-out-of-$n$ systems consisting of arbitrarily dependent components. Under the condition that at least $n-r+1$ components ($r\le n$) of the system are working at time $t$, we study the reliability properties of the residual lifetime of such systems. Also, we present some stochastic ordering properties of residual lifetime of consecutive $k$-out-of-$n$ systems. In the following, we investigate the inactivity time of the component with lifetime $T_{r:n}$ at the system...

A vertex coloring of a graph G is a multiset coloring if the multisets of colors of the neighbors of every two adjacent vertices are different. The minimum k for which G has a multiset k-coloring is the multiset chromatic number χₘ(G) of G. For every graph G, χₘ(G) is bounded above by its chromatic number χ(G). The multiset chromatic numbers of regular graphs are investigated. It is shown that for every pair k, r of integers with 2 ≤ k ≤ r - 1, there exists an r-regular graph with multiset chromatic...

A vertex coloring of a graph $G$ is a multiset coloring if the multisets of colors of the neighbors of every two adjacent vertices are different. The minimum $k$ for which $G$ has a multiset $k$-coloring is the multiset chromatic number ${\chi }_m\left(G\right)$ of $G$. For every graph $G$, ${\chi }_m\left(G\right)$ is bounded above by its chromatic number $\chi \left(G\right)$. The multiset chromatic number is determined for every complete multipartite graph as well as for cycles and their squares, cubes, and fourth powers. It is conjectured that for each $k\ge 3$, there exist sufficiently...

For a nontrivial connected graph $F$, the $F$-degree of a vertex $v$ in a graph $G$ is the number of copies of $F$ in $G$ containing $v$. A graph $G$ is $F$-continuous (or $F$-degree continuous) if the $F$-degrees of every two adjacent vertices of $G$ differ by at most 1. All $P_3$-continuous graphs are determined. It is observed that if $G$ is a nontrivial connected graph that is $F$-continuous for all nontrivial connected graphs $F$, then either $G$ is regular or $G$ is a path. In the case of a 2-connected graph $F$, however, there...