xii List of Figures Fig.5.1 Harmonious and non-harmonious graphs Fig.5.2 Harmonious colorings of graphs. 56 Fig.5.3 Harmonious and non-harmonious graphs Fig.6.1 s of the regions of a map M.(a)M.(b) m 64 Fig.6.2 fthe regions of map M.M. b proper,(c)set,(d)metric,(e)multiset,(f)sum.................. 65 Fig.7.1 A set coloring of a graph..... Fig.7.2 A set 3-coloring of the Grotzsch graph. Fig.7.3 A graph G with x:(G)=1+log,@(G)] 3 Fig.8.1 A multiset 2-coloring of a 4-chromatic graph G. Fig.8.2 The graph H in Case 1 for a=1 and b =4 Fig.8.3 The 2 of the proof of Theorem.1 for a=3 and 二4。。。。。。。 83 Fig.9.1 A4-chromatic graph G with u()=3................. 86 Fig.9.2 A graph G with (G)=3 and (G)=4.......................... Fig.9.3 An 8-chromatic graph G with u(G)=1 [log,@(G)]=4. Fig.9.4 A graph G and a vertex v of G with u(G-v)=u(G)+deg v 9 Fig.9.5 Graphs in the proof of Theorem 9.4.2 for a.10) 92 82 -sigma coloring and a sigma coloring of a graph Fig. %9 Fig.11.1 A 5 x 7 checkerboard... Fig.11.2 A coin placement on the 5 x 7 checkerboard .................. 104 Fig.11.3 A bipartite graph G with mc(G)=3....... 105 Fig.11.4 Modular colorings of C for8≤n≤1i 107 Fig 115 A tree T with mc(T)=3 107 11.6 Lights Out Gar 110 11.7 A bipartite graph G with mc(G)=3. 12 Fig.12. Seating a freshman,a sophomore and a junior 118 Fig.12.2 Iwo seating arrangements of four students ...................... 118 Fig.12.3 Four seating arrangements of five students......................... 118 Fig. 12.4 Seating arrangements of six or more students 119 Fig.12.5 A seating arrangement of 18 students....... 119 Fig.12.6 A seating arrang ment of 36 students 120 xii List of Figures Fig. 5.1 Harmonious and non-harmonious graphs .......................... 54 Fig. 5.2 Harmonious colorings of graphs.................................... 56 Fig. 5.3 Harmonious and non-harmonious graphs .......................... 60 Fig. 6.1 Three colorings of the regions of a map M. (a) M, (b) proper, (c) set, (d) sum .............................................. 64 Fig. 6.2 Four colorings of the regions of a map M. (a) M, (b) proper, (c) set, (d) metric, (e) multiset, (f) sum .................... 65 Fig. 7.1 A set coloring of a graph ............................................ 68 Fig. 7.2 A set 3-coloring of the Grötzsch graph............................. 72 Fig. 7.3 A graph G with s.G/ D 1 C dlog2 !.G/e ........................ 73 Fig. 8.1 A multiset 2-coloring of a 4-chromatic graph G ................... 76 Fig. 8.2 The graph H in Case 1 for a D 1 and b D 4 ....................... 82 Fig. 8.3 The graph G in Case 2 of the proof of Theorem 8.4.1 for a D 3 and b D 4 ................................................. 83 Fig. 9.1 A 4-chromatic graph G with .G/ D 3 ............................ 86 Fig. 9.2 A graph G with .G/ D 3 and !.G/ D 4.......................... 88 Fig. 9.3 An 8-chromatic graph G with .G/ D 1 C dlog2 !.G/e D 4 .... 89 Fig. 9.4 A graph G and a vertex v of G with .G v/ D .G/ C deg v . 91 Fig. 9.5 Graphs in the proof of Theorem 9.4.2 for a 2 f7; 10g and b D 30 ........................................................... 92 Fig. 10.1 A non-sigma coloring and a sigma coloring of a graph ........... 96 Fig. 10.2 3-Colorings of K5.2/;3.3/ ............................................. 99 Fig. 11.1 A 5 7 checkerboard ............................................... 104 Fig. 11.2 A coin placement on the 5 7 checkerboard ...................... 104 Fig. 11.3 A bipartite graph G with mc.G/ D 3 ............................... 105 Fig. 11.4 Modular colorings of Cn for 8  n  11 ........................... 107 Fig. 11.5 A tree T with mc.T/ D 3 ........................................... 107 Fig. 11.6 Lights Out Game .................................................... 110 Fig. 11.7 A bipartite graph G with mc.G/ D 3 ............................... 112 Fig. 12.1 Seating a freshman, a sophomore and a junior..................... 118 Fig. 12.2 Two seating arrangements of four students ........................ 118 Fig. 12.3 Four seating arrangements of five students......................... 118 Fig. 12.4 Seating arrangements of six or more students ..................... 119 Fig. 12.5 A seating arrangement of 18 students .............................. 119 Fig. 12.6 A seating arrangement of 36 students .............................. 120