Group-antimagic Labelings of Multi-cyclic Graphs

Integer-antimagic spectra of disjoint unions of cycles

“…Note that, when n = 1, h = f , where f is defined in Lemma 1.4. Hence I h (C 3 ) =[3,5]. So it still holds.…”

“…Here is a labeling η 3 for C 10 + C 10 = 2C 10 such that I η 3 (2C 10 ) = [3,22]. For C 10 + C 18 , we keep the labeling for the first C 10 in 2C 10 and the labeling for C 18 is:…”

“…If 3 ≤ |G| ≤ 6 and G contains at least two cycles, then G must be 2C 3 . Corollary 5.2 shows that I f 3 (2C 3 ) = [3,9] \ {6} ▹ (5). Thus, the labeling is proper.…”

“…There is a labeling σ3 for G = C 4m+1 + C 4n−1 such that I σ 3 (G) = [3, |G| + 2] ▹ (|G|/2 + 1). Proof.From Lemma 1.4 there are labelings f and g such that I f (C 4m+1 ) = [2, 4m+2]▹(2m+ 1) and I g (C 4n−1 ) = [3, 4n+1]▹(2n+1).…”

“…When (β 1 , δ 1 ) = (0, 1) or(3, 0). Let σ be the labeling of G 1 + H 1 in Case 1 and h be the labeling in Lemma 1.7.…”