ON SUNFLOWER GRAPH, GRID GRAPH AND K1,n+ ¯K2 GRAPH Devi Ayu Lestari, Mania Roswitha, Titin Sri Martini
Department of Mathematics
Faculty of Mathematics and Natural Sciences Sebelas Maret University
Abstract. A simple graph Gadmits an H-magic covering if every edge E belongs to subgraph ofGisomorphic toH. We said graphGisH-magic if there exists total labeling
λ:V(G)∪E(G)→ {1,2, ...,|V(G)|+|E(G)|}, such that for every subraphH′= (V′, E′)
ofGisomorphic to H and satisfyingλ(H′
)def =
∑
v∈V′f(v) +
∑
e∈E′f(e) =m(f), where
m(f) is constant magic sum, then Gis H-supermagic covering if λ(V) ={1,2, ...,|V|}
where s(f) is supermagic sum. In this research we definedC3-supermagic covering on sunflower graphSFnforn≥5 andnodd,C4-supermagic covering on grid graphPn×Pm forn≥6 andm≥6, andC3-supermagic covering onK1,n+ ¯K2graph forn≥3 andnodd.
Keywords: cycle-supermagic covering, sunflower graph, grid graph,K1,n+ ¯K2 graph.
1. Introduction
Labeling is one of the topics in graph theory which is recently examined and developed [2]. A labeling of a graph is a function that carries elements of a graphG to positive or non-negative integers (Wallis [14]). One type of graph labelings which excessively studies by scientists is a magic labeling.
Magic labeling was introduced by Sedl´acˇek [2], then formulated by K¨otzig and Rosa [12]. One of magic labelings that frequently discussed is the edge-magic la-beling. Wallis [14] defined an edge-magic labeling is a one to one mapping from a graph elements, while a set of vertices V(G) and a set of edges E(G) to positive integers{1,2, . . . ,|V(G)|+|E(G)|}, such that for all edgesxy,f(x)+f(y)+f(xy) is constant. An edge magic labeling is called an-edge supermagic labeling if (f(V) = {1,2, . . . ,|V(G)|}).
Edge-magic labeling was developed into a magic covering by Guti´errez and Llad´o [3]. A finite simple graph G admits an H-magic covering if every edge E belongs to subgraph of G isomorphic to H. Then, G is H-magic if there exists a total labeling λ : V(G)∪E(G) → {1,2, ...,|V(G)|+|E(G)|} and magic sum m(f), such that for every subgraph H′ = (V′, E′) of Gisomorphic to H,λ(H′)def
=
∑
v∈V′f(v) +
∑
e∈E′f(e) =m(f). Then, G is H-supermagic covering if λ(V) ={1,2, ...,|V|}.
The research by Guti´errez and Llad´o in [3] provided H-supermagic labeling of stars, complete bipartite graphs Km,n. Then, Llad´o dan Moragas [7] developed a
path-supermagic labeling while Ngurah et al. [11] proved cycle-supermagic labeling of some graphs. Then, Roswitha and Baskoro [13] provedH-magic covering of some graphs. This research aims to determine cycle-supermagic covering on sunflower graph, grid graph, and K1,n+ ¯K2 graph.
2. Main Result
2.1. k-balanced multi-set. Maryati et al. [9] introduced k-balanced multi-set as a tecnique of partitioning a multi-set. We use the following notation. For any two integers a < b, we denote by [a, b], the set of all consecutive integers from a to b, namely{x∈N|a≤x≤b}and the notation∑
Xis for∑
x∈Xx. Here we also define
that {a} ⊎ {a, b}={a, a, b}. Let k ∈N and Y be a multi-set that contains positive
integers. Y is said to bek-balanced if there exists k subsets ofY, say Y1, Y2, . . . , Yk,
such that for every i∈[1, k], |Yi|= |Yk|, ∑
Yi =
∑
Y
k ∈N, and ⊎k
i=1Yi =Y, then Yi
is called a balanced subset of Y.
Later on,k-balanced multi-set is generated to (k;θ)-balanced multi-set by Maryati et al. (see [8]). Let k, θ ∈ N and Y be a multi-set that contains positive integers.
Y is said to be (k, θ)-balanced if there exists k subsets of Y, say Y1, Y2, . . . , Yk,
and θ is distinct numbers, say a1, a2, . . . , aθ, such that ⊎ki=1Y i = Y,|Yi| = |Yk| for
every i ∈ [1, k] and ∑
Yt = ∑
Yt+1 = . . . = ∑Yt−1+k
θ = as for t ∈ {1, k θ + 1,
2k
θ +
1, . . . ,k(θθ−1) + 1}and s= (t−1)θ
k+ 1. For θ = 1, a (k, θ)-balanced is ak-balanced.
Now, we provide some lemmas.
Lemma 2.1. Letm, nbe positive integers, then the multi-set Y = [1, mn] is(n(m− 1),2)-balanced.
Proof. For every i∈[1, n(m−1)] define Yi ={ai, bi} where
ai =
i, 1≤i≤ ⌈n(m2−1)⌉;
mn+⌈n(m2−1)⌉+ 1−i, ⌈n(m2−1)⌉+ 1 ≤i≤n(m−1).
bi =
mn−2−i, 1 ≤i≤ ⌈n(m2−1)⌉;
⌈n
2⌉+i− ⌊
n(m−1)
2 ⌋, ⌈
n(m−1)
2 ⌉+ 1≤i≤n(m−1).
Then, we define the sets
A = {ai,1≤i≤n(m−1)}= [1,⌈n(m−1) 2 ⌉]⊎[⌈
n(m−1)
2 ⌉+n+ 1, mn] B = {bi,1≤i≤n(m−1)}= [⌈
n
Since A⊎B = Y, we have ⊎k
i=1Yi = Y. For every i ∈ [1, n(m−1)], we obtain
|Yi|= 2 and
∑
Yi =
mn−2, 1≤i≤ ⌈n(m2−1)⌉;
mn+⌈n
2⌉+ 2, ⌈
n(m−1)
2 ⌉+ 1 ≤i≤n(m−1).
Then we conclude that Y is (n(m−1),2)-balanced.
Lemma 2.2. Let m, n,andx be positive integers, then the multi-set Z = [x+ 1, x+ (2(m−1)−1)n−⌈m
2⌉+n−2]\{x+n, x+3n−1, x+5n−2, . . . , x+(2(m−1)−1)n−n+2}
is (n−1)-balanced.
Proof. For everyi∈[1,(n−1)], defineZi ={a j
i,1≤j ≤m−1}⊎{b j
i,1≤j ≤m−1}
with
aji = (j−1)(2n−1) +x+i;
bji = x+ (m−1)n+ (n−1)m+ 2−n−i−(j−1)(2n−1).
It can be verified that ⊎k
i=1Zi = Z. Since |Zi| = 2(m −1) and ∑
Zi = (m−
1)(2x+ 2mn−m−2n+ 2) is constant, for every i∈[1,(n−1)]. Thus, we conclude
that Z is (n−1)-balanced.
Lemma 2.3. Let k be an odd integer and x be positive integers, then the multi-set Y = [3, k+ 2]⊎[k+ 4,2k+ 3]⊎[x+ 5−k, x+ 4] is k-balanced.
Proof. For every i∈[1, k], define Yi ={ai, bi, ci} where
ai = 2 +i, i= 1,2, . . . , k.
bi =
1
2(3k+ 7 + 2i), i= 1,2, . . . ,
k−1 2 ; 1
2(k+ 7 + 2i), i=
k+1 2 ,
k+3
2 , . . . , k.
ci =
x+ 5−2i, i= 1,2, . . . ,k−1 2 ;
x+k+ 5−2i, i= k+1 2 ,
k+3
2 , . . . , k.
Let
A = {ai,1≤i≤n}= [3, k+ 2]
B = {bi,1≤i≤n}= [k+ 4,2k+ 3]
Since A⊎B ⊎C = Y, we have ⊎k
i=1Yi = Y. For every i ∈ [1, k] we find that
|Yi|= 3 and ∑
Yi =x+12(3k+ 21) is constant. Therefore,Y is k-balanced.
Here we give two examples as follows.
(1) Let Y = [1,36], m = 6, and n = 6. By Lemma 2.1, for every i∈ [1,30], we have 30-subsets of Y as follows.
Y1 ={1,33}; Y2 ={2,32}; Y3 ={3,31}; Y4 ={4,30};
Y5 ={5,29}; Y6 ={6,28}; Y7 ={7,27}; Y8 ={8,26}; Y9 ={9,25}; Y10={10,24}; Y11={11,23}; Y12={12,22};
Y13={13,21}; Y14={14,20}; Y15={15,19}; Y16={36,4};
Y17={35,5}; Y18={34,6}; Y19={33,7}; Y20={32,8};
Y21={31,9}; Y22={30,10}; Y23={29,11}; Y24={28,12}; Y25={27,13}; Y26={26,14}; Y27={25,15}; Y28={24,16};
Y29={23,17}; Y30={22,18}.
It is easy to check that for everyi∈[1,15],∑
Yi = 34 and fori∈[16,30], ∑
Yi =
40. Hence, Y is (30,2)-balanced.
(2) Let Y = [3,5]⊎[7,9]⊎[11,13], x = 9 and k = 3. By Lemma 2.3, for every i∈[1,3], we have 3-subsets of Y as follows.
Y1 ={3,9,12} Y2 ={4,7,13} Y3 ={5,8,11}
Clearly,∑
Yi = 24, and so Y is 3-balanced.
2.2. (k, δ)-anti balanced multi-set. Inayah et al. in [5] defined (k, δ)-anti bal-anced multi-sets as follows. Let k, δ ∈ N and X be a multi-set containing positive integers. X is said to be (k, δ)-anti balanced if there exist k subsets of X, say X1, X2, X3, . . . , Xk, such that for every i ∈ [1, k],|Xi| = |
X| k ,
⊎k
i=1Xi =X, and for
i∈[1, k−1], ∑
Xi+1−∑Xi =δ is satisfied.
Next, we give several lemmas about (k, δ)-anti balanced multi-sets.
Lemma 2.4. Letk ≥5be an odd integer. IfY = [3k+2,4k+1]thenY is(k,1)-anti balanced.
Proof. For every i∈[1, k],
Yi =
{3k+⌈k
2⌉+ 1− ⌊
i
2⌋,4k+ 2− ⌈
i
2⌉}, if i is odd;
{4k+ 2− i
2,3k+⌈
k
2⌉+ 1−
i
It can be verified that for every i ∈ [1, k], |Yi| = 2, Yi ⊂ Y and ⊎k
i=1Yi = Y.
Since ∑
Yi = 7k+⌈k2⌉+ 3−i for every i∈[1, k], then ∑
(Yi)− ∑
(Yi+1) = 1, Y is
(k,1)-anti balanced.
Lemma 2.5. Letk ≥5be an odd integer. IfX ={1,3,5, . . . ,2k−1}⊎[4k+2,5k+1] then X is(k,1)-anti balanced.
Proof. For every i∈[1, k] we define
Xi =
{i,5k+ 2−i, k+i−1}, if i is odd;
{k+i,5k+ 2−i, i−1}, if i is even.
It is obvious that for each i ∈ [1, k], |Xi| = 3, Xi ⊂ X and ⊎k
i=1Xi = X. Since ∑
Xi = 6k+ 1 +i for every i∈[1, k], then ∑(Xi+1)−∑(Xi) = 1, X is (k,1)-anti
balanced.
Lemma 2.6. Let k ≥ 5 be an odd integer. If P ={2,4,6, . . . ,2k} ⊎[2k+ 2,3k+ 1]⊎[5k+ 2,6k+ 1] then P is(k,1)-anti balanced.
Proof. For every i∈[1, k] wherek(mod 4)≡1 we construct
Pi =
{k+ 2−i,2k+⌈k
2⌉+
i+1
2 ,6k+ 1−
i−1
2 }, if i is odd;
{2k+ 2−i,2k+⌈k
2⌉+ 1−
i
2,5k+ 1 +
i
2}, if i is even.
and for every k(mod 4)≡3,
Pi =
{2k+ 1−i,2k+⌈k
2⌉ −
i−1
2 ,5k+ 2 +
i+1
2 }, if i is odd;
{2k+ 1−i,3k− ⌊k
2⌋+ 1,5k+ 2}, if i=k;
{k+ 1−i,2k+⌈k
2⌉+ 1 +
i
2,6k+ 2−
i
2}, if i is even.
We have that for every i ∈ [1, k], |Pi| = 3, Pi ⊂ P and ⊎k
i=1Pi = P. Since ∑
Pi = 9k +⌈k2⌉ + 4−i for every i ∈ [1, k], then ∑
(Pi) − ∑
(Pi+1) = 1, P is
(k,1)-anti balanced.
2.3. C3-supermagic labeling ofSFn. A sunflower graphSFnis defined as follows:
consider a wheel graphWn with center vertex c and ann-cycle v1, v2, v3, . . . , vn and
additional n vertices w0, w1, w2, . . . , wn−1 where wi is joined by an edge to vi, vi+1
Theorem 2.7. A sunflower graph SFn for n≥5 and n odd is C3-supermagic.
Proof. LetGbe a sunflower graphSFn and letH beC3. Then |V(G)|= 2n+ 1 and
|E(G)|= 4n. We define a bijective functionξ1 :V(G)∪E(G)→ {1,2, . . . ,6n+ 1}.
Let W = [1,6n + 1]. We partition W into four sets Y, X, Z, P such that W =
Y ∪X∪Z∪P withY = [3n+ 2,4n+ 1],X ={1,3,5, . . . ,2n−1} ⊎[4n+ 2,5n+ 1],
Z ={2n+ 1} and P ={2,4,6, . . . ,2n} ⊎[2n+ 2,3n+ 1]⊎[5n+ 2,6n+ 1].
All vertices and edges in each subgraph Hi are labeled by some rules. According
to Lemma 2.5, for k = n, we have X is (n,1)-anti balanced. We use the set X =
{1,3,5, . . . ,2n −1} to label vertex vi. Then label every edges incident to vi with
the elements of interval [4n+ 2,5n+ 1], so we obtain ∑
Xi = 6n+ 1 +i. Label the
center vertex cwith 2n+ 1.
By applying Lemma 2.4, we set k = n. Then, we label the edges vic by the
elements of the interval [3n+ 2,4n+ 1] such that we obtain a magic sum for each
subgraphsYibased on Lemma 2.4. Therefore, Y is (n,1)-anti balanced. Then based
on Lemma 2.6, for k = n we conclude that P is (n,1)-anti balanced. We use the
elements of {2,4,6, . . . ,2n} to label vertex wi−1, then use the elements of interval
[2k+ 2,3k+ 1]⊎
[5k+ 2,6k+ 1] to label edge viwi−1 such that by Lemma 2.6 we
obtain that∑
Pi = 9n+⌈n2⌉+ 4−i. It can be checked thatξ1 is a bijective function
of the elements of V(G)∪E(G) to {1,2,3, . . . ,6n+ 1}. Since H isomorphic with
C3 for every i ∈ [1, k], then f(Hi) is constant. Therefore the sunflower SFn is
C3-supermagic.
2.4. C4-supermagic labeling of Pn×Pm. Ngurah et al. [11] defined a grid graph
Pn×Pm as a graph obtained from product operation between two path graphs Pn
and Pm with mnvertices and (m−1)n+ (n−1)m edges, forn, m≥6.
Theorem 2.8. A grid graph Pn×Pm for n ≥6 and m≥6 is C4-supermagic.
Proof. Let Gbe a grid graph Pn×Pm and let H be a C4. Then |V(G)|=mn and
|E(G)| = (n −1)m+m −1(n). Define a bijective function ξ2 : V(G)∪E(G) →
{1,2, . . . ,3nm−n−m}. Let U = [1,3nm−n−m]. We partition U into two sets,
Based on Lemma 2.1, we obtain Y is (n(m −1),2)-balanced. The elements of
[1, mn] fromY is used to labelvij, which every sum of Yi alternate betweenmn−2
and mn+⌈n
2⌉+ 2. Thus, every subgraph Hi has the sum of vertex label
∑
Y =
2mn+⌈n
2⌉. Then we label the set of edges based on Lemma 2.2. Let E be a set of
edges. Then, we partition E into two setsE1 andE2, such that E =E1∪E2. Label
E1 and E2 based on Lemma 2.2 forx=mnandx=mn+n−1, respectively. Thus,
the sum of all edge labels of every subgraphC4 of Pn×Pm is 8mn−2n−2m+ 2. It
is obvious that ξ2 is a bijective function of V(G)∪E(G) to{1,2, ...,3nm−n−m}.
If H is isomorphic to C4, for every i ∈[1,(m−1)(n−1)], then f(Hi) is constant.
Therefore a grid graph Pn×Pm is C4-supermagic.
2.5. C3-supermagic labeling of K1,n+ ¯K2. A complete bipartite graph Km,n is a graph that can be partitioned into two subsets V1 and V2 where |V1| = m and
|V2| = n (Johnsonbaugh [6]). A Complete graph is a graph with every vertex is
adjacent (Chartrand dan Lesniak [1]). A K1,n + ¯K2 is a graph that obtained from
join operation between a complete bipartite graphK1,n and a ¯K2 withn+ 3 vertices and 3n+ 2 edges.
Theorem 2.9. A K1,n+ ¯K2 graph for n≥3 and n odd is C3-supermagic.
Proof. Let G be a K1,n + ¯K2 graph with |V(G)| = n+ 3 and |E(G)| = 3n + 2.
Let Hi be C3 of G. The number of subgraphs in G that isomorphic to C3 is 2n.
We define a bijective function ξ3 : V(G)∪E(G) → {1,2, . . . ,4n + 5}. Let M =
[1,4n+ 5]. We partition M into two sets Y and K such that M = Y ∪K with
Y = [3, n+ 2]⊎[n+ 4,2n+ 3]⊎[2n+ 5,4n+ 4] andK ={1,2, n+ 3,2n+ 4,4n+ 5}.
Now, we defineξ3 labeling onG. Label the vertex centerx1 with 1, verticesc1 and
c2 with 2 and n+ 3, respectively. Then we label edgesx1c1, x1c2 with 4n+ 5,2n+ 4
respectively. Use Lemma 2.3 to partitionY be {Yi,1≤i≤k}, withYi ={ai, bi, ci}.
Based on Lemma 2.3, for x = 3n and k = n, we get Y is n-balanced. Use ai
from Yi to label vi, bi to label vix1, and ci to label vic1. Thus, we obtain
∑
Yi =
3n+ 12(3n+ 21). Then, use Lemma 2.3 for x = 4n and k = n, therefore we get
Y is n-balanced. Use ai from Yi to label vi, bi to label vix1, and ci to label vic2.
We obtain ∑
Yi = 4n+ 12(3n+ 21). If H is isomorphic to C3 for every i ∈ [1, n],
3. CONCLUSION
We have presented the results of cycle-supermagic on sunflower graph, grid graph andK1,n+ ¯K2 graph. We have found that a sunflower graph isC3-supermagic for n ≥ 5 and n odd (Theorema 2.7), a grid graph is C4-supermagic for n ≥ 6 and
m ≥ 6 (Theorema 2.8) and a K1,n + ¯K2 graph is C3-supermagic for n ≥ 3 and n
odd (Theorema 2.9).
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