Upper bounds for the 2-hued chromatic number of graphs in terms of the independence number

A 2-hued coloring of a graph $G$ (also known as conditional $(k, 2)$-coloring and dynamic coloring) is a coloring such that for every vertex $v\in V(G)$ of degree at least $2$, the neighbors of $v$ receive at least $2$ colors. The smallest integer $k$ such that $G$ has a 2-hued coloring with $k$ colors, is called the {\it 2-hued chromatic number} of $G$ and denoted by $\chi_2(G)$. In this paper, we will show that if $G$ is a regular graph, then $\chi_{2}(G)- \chi(G) \leq 2 \log _{2}(\alpha(G)) +\mathcal{O}(1)$ and if $G$ is a graph and $\delta(G)\geq 2$, then $\chi_{2}(G)- \chi(G) \leq 1+\lceil \sqrt[\delta -1]{4\Delta^{2}} \rceil ( 1+ \log _{\frac{2\Delta(G)}{2\Delta(G)-\delta(G)}} (\alpha(G)) )$ and in general case if $G$ is a graph, then $\chi_{2}(G)- \chi(G) \leq 2+ \min \lbrace \alpha^{\prime}(G),\frac{\alpha(G)+\omega(G)}{2}\rbrace$.Comment: Dynamic chromatic number; conditional (k, 2)-coloring; 2-hued chromatic number; 2-hued coloring; Independence number; Probabilistic metho

Sigma Partitioning: Complexity and Random Graphs

A $\textit{sigma partitioning}$ of a graph $G$ is a partition of the vertices into sets $P_1, \ldots, P_k$ such that for every two adjacent vertices $u$ and $v$ there is an index $i$ such that $u$ and $v$ have different numbers of neighbors in $P_i$. The $\textit{ sigma number}$ of a graph $G$, denoted by $\sigma(G)$, is the minimum number $k$ such that $G$ has a sigma partitioning $P_1, \ldots, P_k$. Also, a $\textit{ lucky labeling}$ of a graph $G$ is a function $\ell :V(G) \rightarrow \mathbb{N}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w)$ ($x \sim y$ means that $x$ and $y$ are adjacent). The $\textit{ lucky number}$ of $G$, denoted by $\eta(G)$, is the minimum number $k$ such that $G$ has a lucky labeling $\ell :V(G) \rightarrow \mathbb{N}_k$. It was conjectured in [Inform. Process. Lett., 112(4):109--112, 2012] that it is $\mathbf{NP}$-complete to decide whether $\eta(G)=2$ for a given 3-regular graph $G$. In this work, we prove this conjecture. Among other results, we give an upper bound of five for the sigma number of a uniformly random graph

Optimal Sensor Deception to Deviate from an Allowed Itinerary

In this work, we study a class of deception planning problems in which an agent aims to alter a security monitoring system's sensor readings so as to disguise its adversarial itinerary as an allowed itinerary in the environment. The adversarial itinerary set and allowed itinerary set are captured by regular languages. To deviate without being detected, we investigate whether there exists a strategy for the agent to alter the sensor readings, with a minimal cost, such that for any of those paths it takes, the system thinks the agent took a path within the allowed itinerary. Our formulation assumes an offline sensor alteration where the agent determines the sensor alteration strategy and implement it, and then carry out any path in its deviation itinerary. We prove that the problem of solving the optimal sensor alteration is NP-hard, by a reduction from the directed multi-cut problem. Further, we present an exact algorithm based on integer linear programming and demonstrate the correctness and the efficacy of the algorithm in case studies

Modeling of Steam Distillation Mechanism during Steam Injection Process Using Artificial Intelligence

Steam distillation as one of the important mechanisms has a great role in oil recovery in thermal methods and so it is important to simulate this process experimentally and theoretically. In this work, the simulation of steam distillation is performed on sixteen sets of crude oil data found in the literature. Artificial intelligence (AI) tools such as artificial neural network (ANN) and also adaptive neurofuzzy interference system (ANFIS) are used in this study as effective methods to simulate the distillate recoveries of these sets of data. Thirteen sets of data were used to train the models and three sets were used to test the models. The developed models are highly compatible with respect to input oil properties and can predict the distillate yield with minimum entry. For showing the performance of the proposed models, simulation of steam distillation is also done using modified Peng-Robinson equation of state. Comparison between the calculated distillates by ANFIS and neural network models and also equation of state-based method indicates that the errors of the ANFIS model for training data and test data sets are lower than those of other methods