Conflict-Free Coloring of Planar Graphs

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number chi_CF(G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N[v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N(v), for which vertex v is not a member of its neighborhood. For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. We also give a complete characterization of the computational complexity of conflict-free coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G, but polynomial for outerplanar graphs. Furthermore, deciding whether chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for outerplanar graphs. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs. For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general} planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on Discrete Mathematics) of extended abstract that appears in Proceeedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pp. 1951-196

Mitigating load burden on smart grid via EVs: A case study on harnessing EVs as mobile battery for society

World energy consumption is quickly rising as a result of population and economic expansion, particularly in big emerging economies, which will account for 90% of energy demand increase through 2035. Electric vehicles (EVs) are critical components of the electrification revolution aimed at reducing the carbon footprint. In this case study, a completely different side of EVs is explored where EVs can be used as an energy storage unit that has the potential to meet the demands of high energy needs in a variable electricity tariff setting. The study proposed in this work suggests that energy stored in EVs can also be used back in the smart grid at the time of high energy requirements which can significantly decrease the load shedding in both urban and rural areas. The simulation model presented on MATLAB shows a significant dip in energy demand after electricity stored in the Electric vehicles is used back in the smart grid. The study also proposed an ensemble model that is able to predict the overload in the Grid. The ensemble model achieving the R2 score of 0.87 and RMSE value of 0.06

Mitigating Load Burden on Smart Grid Via EVs: A Case Study on Harnessing EVs as Mobile Battery for Society

World energy consumption is quickly rising as a result of population and economic expansion, particularly in big emerging economies, which will account for 90% of energy demand increase through 2035. Electric vehicles (EVs) are critical components of the electrification revolution aimed at reducing the carbon footprint. In this case study, a completely different side of EVs is explored where EVs can be used as an energy storage unit that has the potential to meet the demands of high energy needs in a variable electricity tariff setting. The study proposed in this work suggests that energy stored in EVs can also be used back in the smart grid at the time of high energy requirements which can significantly decrease the load shedding in both urban and rural areas. The simulation model presented on MATLAB shows a significant dip in energy demand after electricity stored in the Electric vehicles is used back in the smart grid. The study also proposed an ensemble model that is able to predict the overload in the Grid. The ensemble model achieving the R2 score of 0.87 and RMSE value of 0.06

Conflict-Free Coloring of Graphs

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry and are well studied in graph theory. Here we study the natural problem of the conflict-free chromatic number χCF(G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N[v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N(v), for which vertex v is not a member of its neighborhood. For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K [subscript k+1] as a minor, then χCF(G) ≤ k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. In addition, we give a complete characterization of the algorithmic/computational complexity of conflict-free coloring. It is NP-complete to decide whether a planar graph has a conflict-free coloring with one color, while for outerplanar graphs, this can be decided in polynomial time. Furthermore, it is NP-complete to decide whether a planar graph has a conflict-free coloring with two colors, while for outerplanar graphs, two colors always suffice. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs. For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k ∈ {1,2,3}, it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general planar graph has a conflict-free coloring with at most eight colors. Keywords: conflict-free coloring; planar graphs; complexity; worst-case boun