A Survey on Monochromatic Connections of Graphs

The concept of monochromatic connection of graphs was introduced by Caro and Yuster in 2011. Recently, a lot of results have been published about it. In this survey, we attempt to bring together all the results that dealt with it. We begin with an introduction, and then classify the results into the following categories: monochromatic connection coloring of edge-version, monochromatic connection coloring of vertex-version, monochromatic index, monochromatic connection coloring of total-version.Comment: 26 pages, 3 figure

The number of removable edges in a 4-connected graph

AbstractLet G be a 4-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting the graph G−e; second, for all the vertices x of degree 3 in G−e, delete x from G−e and then completely connect the 3 neighbors of x by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by G⊖e. If G⊖e is still 4-connected, then e is called a removable edge of G. In this paper we prove that every 4-connected graph of order at least six (excluding the 2-cyclic graph of order six) has at least (4|G|+16)/7 removable edges. We also give the structural characterization of 4-connected graphs for which the lower bound is sharp

Sufficient Conditions for the Exact Relaxation of Complementarity Constraints for Storages in Multi-period ACOPF

Storage-concerned Alternative Current Optimal Power Flow (ACOPF) with complementarity constraints is highly non-convex and intractable. In this letter, we first derive two types of relaxation conditions, which guarantee no simultaneous charging and discharging (SCD) in the relaxed multi-period ACOPF. Moreover, we prove that the regions on LMPs formed by the proposed two conditions both contain the other four typical ones. We also generalize the application premise of sufficient conditions from the positive electricity price requirements to the negative electricity price scenarios. The case studies verify the exactness and advantages of the proposed method