On friendly index sets of k-galaxies

Let G = (V,E) be a graph. A vertex labeling f : V → Z2 induces an edge labeling f ∗ : E → Z2 defined by f ∗(xy) = f(x) + f(y), for each edge xy ∈ E. For i ∈ Z2, let vf (i) = |{v ∈ V : f(v) = i}| and ef (i) = |{e ∈ E : f ∗(e) = i}|. We say that f is friendly if |vf (1) − vf (0)| ≤ 1. The friendly index set of G, denoted by FI(G), is defined as FI(G) = {|ef (1) − ef (0)| : vertex labeling f is friendly}. A k-galaxy is a disjoint union of k stars. In this paper, we establish the friendly index sets for various classes of k-galaxies.


Introduction
Let G = (V, E) be a graph.A vertex labeling f : V → Z 2 induces an edge labeling f * : E → Z 2 defined by f * (xy) = f (x) + f (y), for each edge xy ∈ E.
In 1987, Cahit [1] introduced cordial labelings.In the following decades, cordial graph labelings would become a major topic of study.Motivated by this particular type of labeling, the friendly index set FI(G) of a graph G was introduced [3].The set FI(G) is defined as {|e f (0) − e f (1)| : vertex labeling f is friendly}.When the context is clear, we will drop the subscript f .G is cordial if and only if 0 or 1 is in FI(G).
Cairnie and Edwards [2] have determined the computational complexity of cordial labelings.Deciding whether a graph admits a cordial labeling or not is an NP-complete problem.Even the restricted problem of deciding whether a connected graph of diameter two has a cordial labeling is NP-complete.Thus in general, it is difficult to determine the friendly index sets of graphs.
In [7], the friendly index sets of complete bipartite graphs and cycles are determined.In [5,6,8,9,10,11], the friendly index sets of other classes of graphs are determined.For further details of known results on friendly labelings and friendly index sets, the reader is directed to Gallian's [4] comprehensive survey of graph labelings.
To gain insight into a graph labeling problem, one usually begins by focusing on specific classes of graphs.In this paper, we establish the friendly index sets for various disjoint unions of stars.

Galaxies with identical stars
Let n ≥ 1 and St(n) denote the star with n pendant edges.The following result is well-known [11].
A k-galaxy is a disjoint union of k stars.Consider the galaxy St(n [2m] ), the disjoint union of 2m copies of St(n), where m, n ≥ 1.This particular galaxy has 2mn + 2m vertices and 2mn edges.We use the notation ∆e = e(1) − e(0) and ∆v = v(1) − v(0).
Proof.We determine all of the possible values of ∆e.Let k of the centers of the 2m stars be labeled 0. Without loss of generality, let this be the first, second, . . ., and kth star.Let x i be the number of pendant vertices of the ith star that are labeled 0.Then, e (1) Clearly, k ranges from 0 to 2m.However, we may assume that k ranges from 0 to m; otherwise changing all the vertex labels to their complements still leaves a friendly vertex labeling with the same friendly index and (2m − k) centers labeled 0. Thus, all the possible values of ∆e are 2m + 2k(n − 1) Proof.For any integer j, we see that −2(2j + 1) + 2 is divisible by 4.
Proof.It suffices to show that all values of |∆e| (as asserted) are attainable.Partition St(n [2m] ) into m two-star galaxies St(n, n), i.e., m pairs of stars St(n).We give two sets of labelings.
First, for each pair of stars, label one center with 1 and the pendant vertices of this star with 0, and label the other center with 0 and the pendant vertices of this star with 1.Clearly, this vertex labeling is friendly.Furthermore, e(1) = 2mn and e(0) = 0, giving ∆e = 2mn.Interchange the labels of two pendant vertices in the first pair of stars, creating two edges with label 0. This makes e(1) = 2mn − 2, e(0) = 2, and ∆e = 2mn − 4. Continue with other pairs of pendant vertices, and then with other pairs of stars, giving friendly indices 2mn − 4i with i = 0, 1, . . ., mn.
Second, for each (except the last) pair of stars, use the initial labeling as in the previous paragraph.For the last pair of stars, label one center with 0 and the pendant vertices of this star with 1, and label the other center with 0, one pendant vertex of this star with 1 and the other pendant vertices with 0. Clearly, this vertex labeling is friendly.Furthermore, e(1) = 2(m − 1)n + (n + 1) and e(0) = n − 1, giving ∆e = 2(m − 1)n + 2. Interchange the labels of the pendant vertices in each (except the last) pair of stars as in the previous paragraph, giving friendly indices 2mn−2n+2−4i with i = 0, 1, . . ., (m − 1)n.
Proof.It suffices to show that all the values of |∆e| in the lemma are attainable.Partition St(n [2m+1] ) into m two-star galaxies St(n, n), i.e., m pairs of stars St(n), and a single star St(n).Use the initial labeling as in the previous proof for the m two-star galaxies.For the last star, we present two labelings.

General galaxies
In the analysis of general galaxies, we use the known concept of perfect partitions [12].Consider the galaxy St(a 1 , a 2 , . . ., a n ), where n, a 1 , a 2 , . . ., a n ≥ 2.
For the rest of this section (unless we indicate otherwise), we assume that the partition problem for the multiset {b 1 , . . ., b n } has a perfect solution, and we use the above notation.Proof.The second inclusion is obvious.Label the centers of the first k stars and the pendant vertices of the last (n − k) stars with 0, and all other vertices with 1.The vertex labeling is friendly, giving a friendly index of |E|.Interchange the 1-labels on the pendant vertices of the first k stars with the 0-labels on the pendant vertices of the last (n − k) stars, decreasing ∆e be 4 after each interchange.This generates the friendly indices |E| − 4i, where i = 0, 1, . . ., a

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On friendly index sets of k-galaxies | S-M Lee et al.
Theorem 3.2.Let n be even and |E| be odd.
Proof.For (i) and (iii), the second inclusion is obvious.Label the centers of the first k stars and the pendant vertices of the last (n−k) stars with 0, and all other vertices with 1.The vertex labeling is friendly, giving a friendly index of |E|.Interchange the 1-labels on the pendant vertices of the first k stars with the 0-labels on the pendant vertices of the last (n − k) stars, decreasing ∆e by 4 after each interchange.This generates the friendly indices |E| − 4i, where i = 0, 1, . . ., The friendly indices from the above procedure are |E|−4i, where i = 0, 1, . . ., a (iv).This is the case −(a The friendly indices from the above procedure are |E| − 4i, where i = 0, 1, . . ., a k+1 + Theorem 3.3.Suppose that St(a 1 , a 2 , . . ., a n ), where n, a 1 , . . ., a n ≥ 2, a i = 2 for some i, and a j > 2 for some j.Furthermore, suppose that the multiset {a 1 − 1, . . ., a n − 1} has a perfect solution.Then, FI(St(a 1 , a 2 , . . ., a n )) = {|E| − 2i ≥ 0 : i ≥ 0}.
Proof.Rearrange if necessary, and assume that a 1 = 2.There exists m, with 1 First, label the centers of the first m stars and the pendant vertices of the last (n − m) stars with 0, and all other vertices with 1.The vertex labeling is friendly, giving a friendly index of |E|.Interchange the 1-labels on the pendant vertices of the first m stars with the 0-labels on the pendant vertices of the last (n − m) stars, decreasing ∆e by 4 after each interchange.This generates the friendly indices |E| − 4i, where i = 0, 1, . . ., Second, keep the initial labeling above, except to interchange the 0-label on the center of the first star with the 1-label on a pendant vertex of the same star.This gives a friendly index of |E|−2.Interchange the 1-labels of the pendant vertices of the first m stars (except the first one) with the 0labels on the pendant vertices of the last (n − m) stars, decreasing ∆e by 4 after each interchange.This generates the friendly indices |E| − 2 − 4i, where i = 0, 1, . . ., In other words, all non-negative integers that are decrements of 4 from |E| − 2 are attainable friendly indices.
There are |V | = n+a 1 +a 2 +• • •+ a n vertices, and |E| = a 1 +a 2 +• • •+a n edges.For each i = 1, 2, . . ., n, define b i = a i −1.Suppose that the partition problem for the multiset {b 1 , b 2 , . . ., b n } has a perfect solution (i.e.there exists a partition of the multiset into two sub-multisets of sizes k and n − k that have sums differing by at www.ejgta.orgOn friendly index sets of k-galaxies | S-M Lee et al.