xiv List of Figures Fig.4.5 a set irregular dge coloring of 4....... 48 Fig.4.6 Showing that si(K2.2)=3 and si(K33)=si(K4.4)=4............ 49 Fig.5.1 A multiset irregular 2-edge coloring.. 52 Fig.5.2 Showing that s(Ps)=3 and mi(Ps)=2.... 2 Fig.5.3 Multiset irregular edge colorings of connected graphs of order 4 Fig.5.4 Fig.6.1 Minimum prope 62 Fig.6.2 a proper sum edge coloring........ Fig.7.1 A modular 3-edge coloring of a graph........... Fig.7.2 A modular 3-edge coloring of the Petersen graph. 10 Fig.8.1 A strong 5-edge coloring of a graph.... Fig.8.2 s of order 7 with st g.&3 The tre romatic index order 15 with ong c omatic index 4... u graphs for k 8 Fig.8.5 Four proper binomial-colored graphs 86 Fig.8.6 A labeled proper 3-binomial-colored graph Fig.8.7 Illustrating a step of the proof of Theorem 8.5................ 导 Fig.8.8 Illustrating the coloring c in the proof of Theorem 8.6 fork4 Fig.8.9 A graph G with (G)=4.. A graph G with xi(G)=5. A graph G with A twin edge 4 900 F1g.9.4 Illustrating Mo.M1.M2.M3 and F1.F2.F3.F for K8............... 5 Fig.9.5 A twin edge 7-coloring of S5.5....................................11 xiv List of Figures Fig. 4.5 Constructing a set irregular 4-edge coloring of Q3 and a set irregular 5-edge coloring of Q4 ................................. 48 Fig. 4.6 Showing that si.K2;2/ D 3 and si.K3;3/ D si.K4;4/ D 4 ............ 49 Fig. 5.1 A multiset irregular 2-edge coloring ................................. 52 Fig. 5.2 Showing that s.P5/ D 3 and mi.P5/ D 2 ............................ 52 Fig. 5.3 Multiset irregular edge colorings of connected graphs of order 4 .............................................................. 53 Fig. 5.4 A step in the proof of Theorem 5.8 .................................. 56 Fig. 6.1 Minimum proper sum colorings of graphs .......................... 62 Fig. 6.2 A multiset neighbor-distinguishing of a tree that is not a proper sum edge coloring........................................... 65 Fig. 7.1 A modular 3-edge coloring of a graph ............................... 70 Fig. 7.2 A modular 3-edge coloring of the Petersen graph................... 70 Fig. 8.1 A strong 5-edge coloring of a graph ................................. 82 Fig. 8.2 The trees of order 7 with strong chromatic index 3 ................. 83 Fig. 8.3 A graph of order 15 with strong chromatic index 4 ................. 83 Fig. 8.4 The k-binomial graphs for k D 2; 3 .................................. 86 Fig. 8.5 Four proper binomial-colored graphs................................ 86 Fig. 8.6 A labeled proper 3-binomial-colored graph ........................ 87 Fig. 8.7 Illustrating a step of the proof of Theorem 8.5 ...................... 88 Fig. 8.8 Illustrating the coloring c in the proof of Theorem 8.6 for k D 4 .............................................................. 90 Fig. 8.9 A graph G with 0 ns.G/ D 4 .......................................... 91 Fig. 9.1 A graph G with 0 is.G/ D 5 .......................................... 96 Fig. 9.2 A graph G with 0 ps.G/ D 4 .......................................... 99 Fig. 9.3 A twin edge 4-coloring of the Petersen graph ....................... 102 Fig. 9.4 Illustrating M0; M1; M2; M3 and F1; F2; F3; F4 for K8 ............... 105 Fig. 9.5 A twin edge 7-coloring of S5;5 ....................................... 111