### Properly even harmonious labeling of a union of stars

#### Abstract

*f*is defined as an even harmonious labeling on a graph

*G*with

*q*edges if

*f*:

*V*(

*G*)→{0, 1, …, 2

*q*} is an injection and the induced function

*f*

^{*}:

*E*(

*G*)→{0, 2, …, 2(

*q*− 1)} defined by

*f*

^{*}(

*u*

*v*)=

*f*(

*u*)+

*f*(

*v*) (

*m*

*o*

*d*2

*q*) is bijective. A properly even harmonious labeling is an even harmonious labeling in which the codomain of

*f*is {0, 1, …, 2

*q*− 1}, and a strongly harmonious labeling is an even harmonious labeling that also satisfies the additional condition that for any two adjacent vertices with labels

*u*and

*v*, 0 <

*u*+

*v*≤ 2

*q*. In , Gallian and Schoenhard proved that

*S*

_{n1}∪

*S*

_{n2}∪ … ∪

*S*

_{nt}is strongly even harmonious for

*n*

_{1}≥

*n*

_{2}≥ … ≥

*n*

_{t}and

*t*<

*n*

_{1}/2 + 2. In this paper, we begin with the related question “When is the graph of

*k*

*n*-star components,

*G*=

*k*

*S*

_{n}, properly even harmonious?" We conclude that

*k*

*S*

_{n}is properly even harmonious if and only if

*k*is even or

*k*is odd,

*k*> 1, and

*n*≥ 2. We also conclude that

*S*

_{n1}∪

*S*

_{n2}∪ … ∪

*S*

_{nk}is properly even harmonious when

*k*≥ 2,

*n*

_{i}≥ 2 for all

*i*and give some additional results on combinations of star and banana graphs.

#### Keywords

#### Full Text:

PDFDOI: http://dx.doi.org/10.5614/ejgta.2024.12.1.1

#### References

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