Strong vertex-magic and super edge-magic total labelings of the disjoint union of a cycle with 3-cycles

Abstract

For more than half a century, the question of deciding which 2-regular graphs possess an edge-magic total labeling remains unsolved. A particularly difficult case involves those 2-regular graphs with many components isomorphic to C3, the cycle on three vertices. For 2-regular graphs, there is an obvious correspondence between an edge-magic total labeling and a vertex-magic total labeling (VMTL). The VMTL is strong or, it is a SVMTL if the smallest labels are used on the vertices. In this paper it is shown that each of the following disjoint unions has a SVMTL, and therefore a super edge-magic total labeling: for t≥4, C6∪(2t−1)C3, C8∪(2t−1)C3, C10∪(2t−1)C3 and 2C5∪(2t−1)C3; for t≥3, C9∪2tC3; for t≥2, C11∪2tC3. Combined with previous work, this now means that each disjoint union Cm∪sC3 of odd order possesses a SVMTL for each m such that 3≤m≤11, with the exception of three known graphs: C4∪C3, C5∪2C3 and C4∪3C3. This supports the conjecture stating that these are the only odd-order 2-regular graphs without a SVMTL. It also results in further evidence supporting MacDougall's conjecture regarding which regular graphs possess VMTLs, in general.It is a noteworthy feature of the proofs that a single Kotzig array is used to prove multiple theorems, providing a unifying element to labelings of different types of 2-regular graphs. The proofs are constructive, and this work is self-contained.