List of Figures Fig.2.1 Showing s(K3)=3.. 1 Fig.2.2 An edge coloring of the Petersen graph............................. 9 Fig.2.3 An edge coloring of K4...... 12 Fig.2.4 An edge coloring of Ka Fig.2.5 Constructing the graph H inK 公 g2.6 the proof of for 044.444.4444444.4“444444。小。 Fig.2.7 Edge cole 0ngs0 f Co and C13....... Fig.2.8 Edge c0l0mngs0fC10andC12.......................... 19201 Fig.2.9 Edge colorings of C and Cu.................. Fig.2.10 Illustrating that the inequality in Proposition 2.14 can be strict.... Fig.2.11 Illustrating the equality in Proposition 2.14........ 23 Fig.2.12 An edge coloring of the tree T Fig.2.13 An edge coloring graph 21 Fg2.14 An edge coloring of a connected graph of Fig.3.1 Illustrating B-valuations of C3 and C.. 32 Fig.3.2 Three graphs that are not graceful..... Fig.3.3 Two modular edge-graceful graphs and a non-modular edge-graceful graph........ Fig.3.4 Two modular edge-graceful trees... Fig.3.5 The colorings in Subease 2.I fors3 ands7. 34 Fig.3.6 Two modular edge-graceful colorings of a graph.. 号 The graph set irregular chromatic index Set irregular edge colorings of K and K....... Fig.4.3 A set irregular 4-edge coloring of Ks.. Fig.4.4 A set irregular 3-edge coloring of 2=C4......................... 41List of Figures Fig. 2.1 Showing s.K3/ D 3 ................................................... 7 Fig. 2.2 An edge coloring of the Petersen graph ............................. 9 Fig. 2.3 An edge coloring of K4;4 ............................................. 12 Fig. 2.4 An edge coloring of K5;5 ............................................. 14 Fig. 2.5 Constructing the graph H in K3.4/ .................................... 15 Fig. 2.6 Edge colorings of Pn in the proof of Theorem 2.12 for 19 Fig. 2.7 Edge colorings of C9 and C13 ........................................ 20 Fig. 2.8 Edge colorings of C10 and C12 ....................................... 21 Fig. 2.9 Edge colorings of C7 and C11 ........................................ 22 Fig. 2.10 Illustrating that the inequality in Proposition 2.14 can be strict .... 23 Fig. 2.11 Illustrating the equality in Proposition 2.14 ......................... 23 Fig. 2.12 An edge coloring of the tree T2 ...................................... 25 Fig. 2.13 An edge coloring of a unicyclic graph G ........................... 27 Fig. 2.14 An edge coloring of a connected graph of size n C 1 .............. 28 Fig. 3.1 Illustrating ˇ-valuations of C3 and C4 ............................... 32 Fig. 3.2 Three graphs that are not graceful ................................... 32 Fig. 3.3 Two modular edge-graceful graphs and a non-modular edge-graceful graph................................................... 33 Fig. 3.4 Two modular edge-graceful trees .................................... 34 Fig. 3.5 The colorings in Subcase 2:1 for s D 3 and s D 7 .................. 39 Fig. 3.6 Two modular edge-graceful colorings of a graph ................... 41 Fig. 4.1 The graph G7 of order 7 with set irregular chromatic index 3 ...... 44 Fig. 4.2 Set irregular edge colorings of K3 and K4 ........................... 46 Fig. 4.3 A set irregular 4-edge coloring of K8 ................................ 46 Fig. 4.4 A set irregular 3-edge coloring of Q2 D C4 ......................... 47 xiii 6 n 10 ............................................................ 19