Contents 1 Introduction...... 1.1 The Origin of Edge Colorings........................ 1.2 Proper Edge Colorings.... 2 1.3 Proper Vertex Colorings 3 2 The Irregularity Strength of a Graph 5 2.1 Defined ity Strength. 2.2 nregularity Strength of Paths and Cycles 1 2.4 Additional Bounds for the Irregularity Strength of a Graph.......... 2 3 Modular Sum-Defined Irregular Colorings................ 3.1 Graceful Graphs..... 5 3.2 Modular Edge-Graceful Graphs 3.3 Non-modular Edge-Graceful Graphs 3.4 Nowhere-Zero Modular Edge-Gra ceful Graphs 40 4 Set-Defined Irregular Colorings 4.1 The Set Irregular Chromatic Index........... 4.2 Complete Graphs and Hypercubes. 4.3 Complete Bipartite Graphs. 49 5 Multiset-Defined Irregular Colorings 5.1 The Multiset Irregular Chromatic Index....... 少 5.2 Regular Graphs 530 omplete Bipartite Graphs t44t4小t4ttt44tte 55 Max-Min Value Problcms 00t。tttt0t4e000。tet。0t。tt500tttt。t0t。ttt。tt0t+。 5638 6 Sum-Defined Neighbor-Distinguishing Colorings........................ 0 6.1 The Sum Distinguishing Index....................... 6.2 The 1-2-3 Conjecture.. 63 6.3 The Multiset Distinguishing Index........ xiContents 1 Introduction .................................................................. 1 1.1 The Origin of Edge Colorings .......................................... 1 1.2 Proper Edge Colorings.................................................. 2 1.3 Proper Vertex Colorings ................................................ 3 2 The Irregularity Strength of a Graph ..................................... 5 2.1 Sum-Defined Vertex Colorings: Irregularity Strength ................. 5 2.2 On the Irregularity Strength of Regular Graphs ....................... 8 2.3 The Irregularity Strength of Paths and Cycles ......................... 17 2.4 Additional Bounds for the Irregularity Strength of a Graph .......... 22 3 Modular Sum-Defined Irregular Colorings............................... 31 3.1 Graceful Graphs......................................................... 32 3.2 Modular Edge-Graceful Graphs ........................................ 33 3.3 Non-modular Edge-Graceful Graphs................................... 37 3.4 Nowhere-Zero Modular Edge-Graceful Graphs ....................... 40 4 Set-Defined Irregular Colorings............................................ 43 4.1 The Set Irregular Chromatic Index ..................................... 43 4.2 Complete Graphs and Hypercubes ..................................... 45 4.3 Complete Bipartite Graphs ............................................. 49 5 Multiset-Defined Irregular Colorings...................................... 51 5.1 The Multiset Irregular Chromatic Index ............................... 51 5.2 Regular Graphs.......................................................... 54 5.3 Complete Bipartite Graphs ............................................. 55 5.4 Trees ..................................................................... 56 5.5 Max-Min Value Problems .............................................. 58 6 Sum-Defined Neighbor-Distinguishing Colorings ........................ 61 6.1 The Sum Distinguishing Index ......................................... 61 6.2 The 1-2-3 Conjecture ................................................... 63 6.3 The Multiset Distinguishing Index ..................................... 65 xi