Contents 1 Introduction. 1.1 Graph Colorings.......................... 1.2 Proper Vertex Colorings. 2 1.3 Proper Edge Colorings. 4 1.4 5 1.5 ATheorem fro m Dise 3““+440”4。+44+44”44““4+0+4 5 2 Binomial Edge Colorings. 1 2. Strong Edge Colorings.... 2.2 Proper k-Binomial-Colorable Graphs.... 9 2.3 Unrestricted k-Binomial-Colorable Graphs..................... 14 3 Kaleidoscopic Edge Colorings. 19 3.1 Introduction 19 3 nplete Kaleidosco 13 3-Kaleidosc of M Order 34 Maiestic Edge Colorings. 32 Graceful Vertex Colorings. 35 41 Graceful Labelings.… 4. The Graceful Chromatic Number of a Graph....................... 36 4.3 Graceful Chromatic Numbers of Some Well-Known Graphs...... % 4.4 The Graceful Chromatic Numbers of Trees.. Harmonious Vertex Colorings.. 1 1ng5t+tte…4tt4t。+4。 5 armonio 53 Colorings… Harmonic Colorings....... 3509 6 A Map Coloring Problem.... 6.1 A New Look at Map Colorings................... 7 Set Colorings 67 7.1 Set Chromatic Number. ixContents 1 Introduction ................................................................. 1 1.1 Graph Colorings...................................................... 1 1.2 Proper Vertex Colorings ............................................. 2 1.3 Proper Edge Colorings ............................................... 4 1.4 Eulerian Graphs and Digraphs....................................... 5 1.5 A Theorem from Discrete Mathematics............................. 5 2 Binomial Edge Colorings .................................................. 7 2.1 Strong Edge Colorings ............................................... 7 2.2 Proper k-Binomial-Colorable Graphs ............................... 9 2.3 Unrestricted k-Binomial-Colorable Graphs ......................... 14 3 Kaleidoscopic Edge Colorings............................................. 19 3.1 Introduction........................................................... 19 3.2 Complete Kaleidoscopes............................................. 21 3.3 3-Kaleidoscopes of Maximum Order................................ 27 3.4 Majestic Edge Colorings............................................. 32 4 Graceful Vertex Colorings................................................. 35 4.1 Graceful Labelings ................................................... 35 4.2 The Graceful Chromatic Number of a Graph ....................... 36 4.3 Graceful Chromatic Numbers of Some Well-Known Graphs...... 40 4.4 The Graceful Chromatic Numbers of Trees......................... 45 5 Harmonious Vertex Colorings............................................. 53 5.1 Harmonious Labelings ............................................... 53 5.2 Harmonious Colorings ............................................... 55 5.3 Harmonic Colorings.................................................. 59 6 A Map Coloring Problem .................................................. 63 6.1 A New Look at Map Colorings...................................... 63 7 Set Colorings ................................................................ 67 7.1 Set Chromatic Number............................................... 67 ix