List of Figures Fig.1.1 An Eulerian digraph D........ 5 Fig.2.1 A strong 5-edge coloring of a graph G........................ 8 Fig.2.2 A graph of order 16 with strong chromatic index 4. Fig.2.3 Eight k-binomial graphs for=2.3....... 4 r binomial- olorable a 25 dproper 3-bim ustra able grap 3 3 Fg2.7 ing a step of the proof f of Theorem 2. Illustrating the coloring c in the proof o Theorem 2.3.1 for =4.................. 16 Fig.2.8 The structure of a(k+1)-regular k-binomial-colorable graph G of order 2-I in Theorem 2.3.3............ 年 Fg.3.1 A 6-regular 3-kaleidoscope G of order 8 Fig.3.2 Dlustrating a 10-kaleidos scopic coloring ofK in the m355 3 Fig.3.3 each of K7 and K F1g.3.4 An irregular factorization ={F1F2,F3}0f0..。... 42 Fig.3.5 An irregular factorization F2.F of Ku..................... Fig.3.6 The location of the vertices of the graph G6...................... Fig.3.7 The subgraph F2 for r=6 and r=8............................. Fig.3.8 The subgraphs F1.F2 and F:in G6 Fig.3.9 The triangular sets X for Gio and Gs Fig.3.10 sets X1 and X2 for Gs.... Fig.3.11 edges in Gs 32 Fig.4.1 Graceful colorings of Ka and C. 37 Fig.4.2 A graceful 5-coloring of 3... Fig.4.3 Graceful colorings of w6.W7.Ws............................ 41 Fig.4.4 A tree To with (To)=A(To)+3............................ Fig.4.5 A graph G withe(G=「List of Figures Fig. 1.1 An Eulerian digraph D .............................................. 5 Fig. 2.1 A strong 5-edge coloring of a graph G ............................. 8 Fig. 2.2 A graph of order 16 with strong chromatic index 4................ 9 Fig. 2.3 Eight k-binomial graphs for k D 2; 3 ............................... 10 Fig. 2.4 Six proper binomial-colorable graphs .............................. 11 Fig. 2.5 A labeled proper 3-binomial-colorable graph ..................... 12 Fig. 2.6 Illustrating a step of the proof of Theorem 2.2.2................... 13 Fig. 2.7 Illustrating the coloring c in the proof of Theorem 2.3.1 for k D 4 ............................................ 16 Fig. 2.8 The structure of a .kC1/-regular k-binomial-colorable graph G of order 2k 1 in Theorem 2.3.3 ......................... 18 Fig. 3.1 A 6-regular 3-kaleidoscope G of order 8 ........................... 20 Fig. 3.2 Illustrating a 10-kaleidoscopic coloring of K13 in the proof of Theorem 3.2.2.............................................. 23 Fig. 3.3 A 3-kaleidoscopic coloring for each of K7 and K8 ................. 24 Fig. 3.4 An irregular factorization F D fF1; F2; F3g of K10 ............... 25 Fig. 3.5 An irregular factorization fF1; F2; F3g of K11 ...................... 27 Fig. 3.6 The location of the vertices of the graph G6 ....................... 28 Fig. 3.7 The subgraph F2 for r D 6 and r D 8 .............................. 29 Fig. 3.8 The subgraphs F1, F2 and F3 in G6 ................................. 30 Fig. 3.9 The triangular sets X` for G10 and G8 .............................. 30 Fig. 3.10 Edges in the two triangular sets X1 and X2 for G8 ................. 31 Fig. 3.11 Vertical and slanted edges in G8 .................................... 32 Fig. 4.1 Graceful colorings of K4 and C4 .................................... 37 Fig. 4.2 A graceful 5-coloring of Q3 ......................................... 38 Fig. 4.3 Graceful colorings of W6; W7; W8 .................................. 41 Fig. 4.4 A tree T0 with g.T0/ D .T0/ C 3 ................................ 46 Fig. 4.5 A graph G with g.G/ D ˙5ı 3 ..................................... 51 xi