vi Preface Chapter 4 also deals with unrestricted edge coloringsc of graphs but here the induced vertex coloring is defined so that the color c(v)of a vertex v is the set of colors of its incident edges.In Chap.5,the emphasis changes from vertex colorings that are set-defined to those that are multiset-defined.In both cases.the induced vertex colorings c'are vertex-distinguishing. In Chap.6.unrestricted edge colorings c of graphs are once again considered but in this case the induced vertex coloringscare neighbor-distinguishing,that is (≠c'(w)fore ery two adiacent vertices u nd v.In this ch apter. d hoth re olors bel ng is sum-defined and the other where)is mul Itiset-defined.Chapter 7is devoted to unrestricted edge colorings of graphs whose colors are elements of Z of integers modulo k that induce a sum-defined,neighbor-distinguishing vertex coloring. In Chap.8.both proper and unrestricted edge colorings are considered,and the vertex colorings are set-defined,using elements of [k]as colors.In Chap.9.the edge colorings are proper and the vertex colorings considered are sum-defined,using elements of[as colors.In these two chapters,the properties of being vertex. distinguishi and neighbor-distinguishing ar both des bed.Cha nds with edge colo which are proper edge colorin use the t ar sum-defined The following table summarizes all types of edge colorings considered in this book and the resulting vertex colorings.In particular,the table describes,in each chapter: 1.the condition placed on the edge coloring 2 the sets from thedkinion which the edge colors are selected. f the colors,and Chapter 1:Introduction Chapter 2:The Irregularity Strength of a Graph Unrestricted Edge Colorings.N.Sum-defined.Vertex-Distinguishing Chapter 3:Modular Sum-defined,Irregular Colorings ined,Vertex-Distinguishing Chapter 4:Se -Defined Irregula Unrestricted Edg ge Colorings.N.Set-definec Vertex-Distinguishing. Chapter 5:Multiset-Defined Irregular Colorings Unrestricted Edge Colorings.N.Multiset-defined.Vertex-Distinguishing Chapter 6:Sum-Defined Neighbor-Distinguishing Colorings Unrestricted Edge Colorings.N.Sum-defined,Neighbor-Distinguishing. Chapter 7:Modular Sum-Defined Neighbor-Distinguishing Colorings Unrestricted Edge Colorings.Sum-defined,Neighbo Distinguishing vi Preface Chapter 4 also deals with unrestricted edge colorings c of graphs but here the induced vertex coloring is defined so that the color c0 .v/ of a vertex v is the set of colors of its incident edges. In Chap. 5, the emphasis changes from vertex colorings that are set-defined to those that are multiset-defined. In both cases, the induced vertex colorings c0 are vertex-distinguishing. In Chap. 6, unrestricted edge colorings c of graphs are once again considered but in this case the induced vertex colorings c0 are neighbor-distinguishing, that is, c0 .u/ ¤ c0 .v/ for every two adjacent vertices u and v. In this chapter, two vertex colorings c are defined, both where the colors belong to a set Œk, one where c0 .v/ is sum-defined and the other where c0 .v/ is multiset-defined. Chapter 7 is devoted to unrestricted edge colorings of graphs whose colors are elements of Zk of integers modulo k that induce a sum-defined, neighbor-distinguishing vertex coloring. In Chap. 8, both proper and unrestricted edge colorings are considered, and the vertex colorings are set-defined, using elements of Œk as colors. In Chap. 9, the edge colorings are proper and the vertex colorings considered are sum-defined, using elements of Œk as colors. In these two chapters, the properties of being vertex￾distinguishing and neighbor-distinguishing are both described. Chapter 9 ends with a discussion of so-called twin edge colorings, which are proper edge colorings that use the elements of Zk as colors and that induce proper vertex colorings that are sum-defined. The following table summarizes all types of edge colorings considered in this book and the resulting vertex colorings. In particular, the table describes, in each chapter: 1. the condition placed on the edge coloring, 2. the sets from which the edge colors are selected, 3. the definition of the vertex colors, and 4. the property required of the resulting vertex coloring. Chapter 1: Introduction Chapter 2: The Irregularity Strength of a Graph Unrestricted Edge Colorings, N, Sum-defined, Vertex-Distinguishing. Chapter 3: Modular Sum-defined, Irregular Colorings Unrestricted Edge Colorings, Zk, Sum-defined, Vertex-Distinguishing. Chapter 4: Set-Defined Irregular Colorings Unrestricted Edge Colorings, N, Set-defined, Vertex-Distinguishing. Chapter 5: Multiset-Defined Irregular Colorings Unrestricted Edge Colorings, N, Multiset-defined, Vertex-Distinguishing. Chapter 6: Sum-Defined Neighbor-Distinguishing Colorings Unrestricted Edge Colorings, N, Sum-defined, Neighbor-Distinguishing. Chapter 7: Modular Sum-Defined Neighbor-Distinguishing Colorings Unrestricted Edge Colorings, Zk, Sum-defined, Neighbor-Distinguishing.