CC BY
3URAL MATHEMATICAL JOURNAL, Vol. 10, No. 1, 2024, pp. 123-135
DOI: 10.15826/umj.2024.1.011
ALPHA LABELINGS OF DISJOINT UNION OF HAIRY CYCLES
G. Rajasekaranf and L. Uma
Department of Mathematics, Vellore Institute of Technology, Vellore - 632014, Tamil Nadu, India
t raj asekaran.ganapathy@vit .ac.in
Abstract: In this paper, we prove the following results: (1) the disjoint union of n > 2 isomorphic copies of a graph obtained by adding a pendant edge to each vertex of a cycle of order 4 admits an a-valuation; (2) the disjoint union of two isomorphic copies of a graph obtained by adding n > 1 pendant edges to each vertex of a cycle of order 4 admits an a-valuation; (3) the disjoint union of two isomorphic copies of a graph obtained by adding a pendant edge to each vertex of a cycle of order 4m admits an a-valuation; (4) the disjoint union of two nonisomorphic copies of a graph obtained by adding a pendant edge to each vertex of cycles of order 4m and 4m — 2 admits an a-valuation; (5) the disjoint union of two isomorphic copies of a graph obtained by adding a pendant edge to each vertex of a cycle of order 4m — 1 (4m + 2) admits a graceful valuation (an a-valuation), respectively.
Keywords: Hairy cycles, Graceful valuation, a-valuation.
1. Introduction
Notation and terminology not defined here can be found in [2]. Throughout this paper, we denote by Sn and Cn a star on n + 1 vertices and a cycle on n vertices, respectively.
If a labeling f on a graph G with p edges is a one-to-one function from the set of vertices of G to the set {0,1,... ,p} such that, for p pairs of adjacent vertices x and y, the values |f (x) — f (y)| are distinct, then f is called a graceful valuation (a ^-labeling or a ^-valuation) of G. If, in addition, there exists an integer I such that, for each edge xy € E(G), one of the values f (x) and f (y) does not exceed I and the other is strictly greater than I, then the labeling f is called an a-valuation of G with critical value I. Note that a graph with an a-valuation is necessarily bipartite. As a result, such I must be smaller than the smallest of the two vertex labels that yield the edge labeled 1. Let {A, B} be stable sets (a partition) of vertices with x € A and y € B. Without loss of generality, assume that
A = {x € V(G) : f (x) < I}, B = {y € V(G) : f (y) > I}.
Clearly, every a-valuation is also a graceful labeling but not conversely. Rosa pioneered in 1966 [21] the concept of graph ^-labeling. He also presented certain types of vertex labeling as an important tool for decomposing the complete graph K2p+1 into graphs with p edges.
Theorem 1 [21]. Let a graph G with p edges has an a-valuation. Then, for s € N, there exists a G-decomposition of the complete graph K2ps+1.
Specifically, ^-valuations were developed to challenge Ringel's conjecture [19] that K2n+1 can be decomposed into 2n + 1 subgraphs that are all isomorphic to a given tree with n edges. More results about graph labeling are collected and updated regularly in the survey by Gallian [9].
The disjoint union of graphs H1 = (V1 ,E1),H2 = (V2,E2),...,Hn = (Vn, En) is a graph H = H1UH2U ■ ■ -UHn with vertex set V = V1 U V2 U ■ ■ ■ U Vn and edge set E = E1UE2U ■ ■ -UEn,
where V1 n V2 n ■ ■ ■ n Vm = 0. Lakshmi and Vangipuram [13] proved that there is an a-valuation for the quadratic graph Q(4,4k) consisting of four cycles of length 4k, k > 1. Abrham and Kotzig [1] proved that CmUCn has an a-valuation if and only if both m and n are even and m+n = 0 (mod 4). Eshghi and Carter [6] showed several families of graphs of the form C4ni U C4n2 U ■ ■ ■ U C4 that have a-valuations.
The cartesian product GDH of two graphs G and H is the graph with the vertex set
V (GDH) = V (G)DV (H) and the edge set E(GDH) satisfying the following condition:
(xi,x2)(yi,y2) e E(GDH)
if and only if either x1 = y1 and x2y2 e E(H) or x2 = y2 and x1y1 e E(G).
The corona [7] of two graphs H1 and H2, denoted by H1 0 H2, is the graph obtained by taking one copy of H1, which has m vertices, and m copies of H2, and then joining the kth vertex of H1 with an edge to every vertex in the kth copy of H2.
A unicyclic graph H (other than a cycle) is called a hairy cycle if the deletion of any edge e from the cycle of H results in a caterpillar. Thus, the coronas Cn 0 mK1 are examples of hairy cycles. Kumar et. al. [11, 12, 15-18] proved that the hairy cycle Cn 0 K1, n = 0 (mod 4), and graphs obtained by joining two graceful cycles by a path admit a-valuations. They also discussed that the subdivision of a cycle and pendant edges of CnDK4, joining two isomorphic copies of CnDK4, Cn 0 K1, n = 0 (mod 4), and Cn 0 K1, n = 3 (mod 4), are graceful. Moreover, they proved that Cn 0 rK1, n = 3 (mod 4), and Cn 0 K1,n = 0 (mod 4), are k-graceful. Graf [10] established that Cn 0 K1 has a graceful valuation if n = 3 or 4 (mod 8).
Barrientos [3, 4] showed that if G is a graceful graph with order greater than its size, then the graphs G 0 nK1 and G + nK1 are graceful. He also proved that helms (graphs obtained from a wheel by attaching one pendant edge to each vertex) are graceful. Minion and Barrientos, in [5] and [14], studied the gracefulness of G U Pm and Cr U Gn, where Gn is a caterpillar of size n. Frucht and Salinas [8] analyzed the gracefulness of Cm U Pn, n > 3. Ropp [20] showed that the graph (CmDP2) 0 K1 is graceful. Truszczynski [22] conjectured that all unicyclic graphs except the cycle Cn, n = 1 or 2 (mod 4), are graceful.
Labeled graphs are helpful mathematical models for coding theory, such as designing optimal radar, synch-set, missile guidance, and convolution codes with high auto-correlation. They make it easier to perform optimal nonstandard integer encoding.
This study focuses on graceful and a-valuation of some disconnected graphs. The concept of graceful and a-valuation in graph theory has attracted attention from many researchers during the past three decades. The earlier studies motivated us to research the problem that the disjoint union of various hairy cycles C^l U C^ U ■ ■ ■ U C^k admits an a-valuation, which we partially solve in the present paper.
2. Results
Theorem 2. Let G be the graph obtained by the disjoint union of n isomorphic copies of the hairy cycle Cf1. Then, G admits an a-valuation with exactly one missing number p = 4n — 2 and the critical value a = 4n.
Proof. Let n € N, and let Gk, 1 < k < n, be the kth part of G. Let and , where i = 1,2,3,4 and k = 1,2,...,n, denote vertices of the cycle and leaves of the kth part of G, respectively. Clearly, |V(G)| = |E(G)| = 8n.
To define 9 : V(G) — {0,1,2,..., 8n}, we label the vertices of G1 as follows:
9(w1) = 0, 9(w1)=8n — 1, 9(w1) = 3, 9(w1)=8n — 3, 9(x1) = 8n, 9(x2) = 1, 9(x3) = 8n — 2, 9(x4) = 4.
Next, we label the vertices of the remaining parts of G, 2 < k < n, as follows:
9(w3) = ! 4k + 3(i — 3)/2 if i = 1, 3,
9(w) \ 4(2n — k + 1) — i/2 if i = 2,4,
f 4(k + 1) — 5i/2 if i = 2, 4,
9(r2) =
9(x) \ 8n — 4k + (11 — 3i)/2 if i = 1,3 Define the edge labeling f* on E(G1) by
f *(wx) = |9(w) — 9(x)|
for wx € E(G) as follows:
f *(w1w1) = 8n — 1, f *(w1w1) = 8n — 4, f *(w1w1) = 8n — 6, f*(w1w1) = 8n — 3, f*(w1x1) = 8n, f*(w1x2) = 8n — 2, f*(w1x1) = 8n — 5, f*(w1x1) = 8n — 7.
We label the remaining edges of G as follows:
f *(wk wk+1) = 8(n — k + 1) — 2i for i = 1,3, f *(w2fcw3fc) = 8(n — k) + 3, f *(w4fc w2) = 8(n — k) + 5, f *(w3 x3) = 8(n — k) + 10 — 3i for i = 1,2,3, f *(w2 x2) = 8(n — k + 1).
It is clear that all the vertex and edge labels are distinct. Therefore, the graph G is graceful. Next, we prove that the graceful function 9 is an a-valuation with the missing number p = 4n — 2 and the critical value a = 4n. Since the vertex set V of G is partitioned into two sets, V = A U B, we have
A = {0,1, 3, 4, 5, 7, 8, 2, 9,11,12, 6,... , 4n — 3, 4n — 1, 4n, 4n — 6}, B = {8n, 8n — 1, 8n — 2, 8n — 3,..., 4n + 1}.
Clearly, A and B are independent sets. The number a = 4n satisfies f*(w) < a < f*(x) for every ordered pair (w, x) € A x B. Therefore, 9 is an a-valuation of G (see CSl U CSl U CSl in Fig. 1). □
Theorem 3. The disjoint union of two isomorphic copies of CSn admits an a-valuation with exactly one missing number p = 4(n + 1) and the critical value a = 4n + 5.
Proof. Let i = 1,2,3,4 and j = 1,2,..., n. Denote by u (vj) and Uj (vj) the vertices of the cycle and leaves, respectively, in the first and second copies of CSn, respectively. Clearly,
ySn || /""^n^l _ I TP (S~iSn ||
|V(CSn U CSn)| = |E(C4Sn U C4Sn)| = 8(n + 1).
Define tf : V(C£n U C£n) — {0,1,2,..., 8(n + 1)} as follows: we label the vertices of the cycle of the first copy of C4 n by
tf (U1) = 0, tf (U2) = 7n + 8, tf (us) = n + 2, tf (U4) = 6n + 7
11
Figure 1. An a-valuation of Cf1 u Cf1 u Cf1.
and the vertices of the cycle of the second copy of C4 " by
tf(v!) = 6(n + 1), tf(v2) = 3(n + 1), tf(v3) = 5(n + 1), tf(v4) = 4n + 5, respectively. Label the remaining vertices of the leaves in the graph Cfn U Cfn as follows:
tf(Uik) =
8(n + 1) - (k - 1) -
, (n + 2)(i-2) k+ 2
(n + 1)(i - 1)
if i = 1,3, if i = 2,4,
tf(V1k) = 2(n +1), tf(v2k)=6(n +1) - k
for 1 < k < n, and
tf (v3k) = 3n + k + 4, tf (v4k) = 5(n + 1) - k for 1 < k < n,
tf(V3n) = n +1, tf (V4n) = 3n + 4.
It can be verified that all vertices of the graph are labeled and the labels are distinct. Now, we construct labels for the edge set E of the graph. Define a labeling f on E(C4S" U C4Sn) by f(uv) = |tf(u) - tf(v)| for uv e E. We label the edges of the cycle of the first copy of Cf" by
f (u1u2) = 7n + 8, f (u2u3) = 6(n + 1), f (u3u4) = 5(n + 1), f (u4u1) = 6n + 7
and the edges of the cycle of the second copy of C4 n by
f (V1V2) = 3(n + 1), f (V2V3) = 2(n + 1), f (V3V4) = n, f (V4V1) = 2n + 1.
Label the remaining edges of the leaves in the graph C4n U C4n as follows:
for 1 < k < n,
f (UiUik ) =
f(vivik) =
8(n + 1) - (k - 1) - (n + 1)(i - 1) if i = 1,2, 8(n + 1) - k - (n + 1)(i - 1) if i = 3,4,
4(n + 1) - k - (n + 1)(i - 1) if i = 1,2,
4(n + 1) - k - 1 - (n + 1)(i - 1) if i = 3, 4,
2
for 1 < k < n, and
f (V3V3n) = 4(n + 1), f (V4V4n )= n + 1.
It is quite clear that all the vertex and edge labels are distinct. Therefore, the graph C4n U C4n is graceful. Next, we prove that this graceful function tf is an a-valuation with the missing number p = 4(n + 1) and the critical value a = 4n + 5. Since the vertex set V of Cfn U Cfn is partitioned into two sets, V = R U S, we have
R = {0,1, 2,... , n, n + 2, n + 3,..., 2(n + 1), 6(n + 1), 6n + 5, 6n + 4, ..., 5n + 6, 2(2n + 3), 4n + 5,..., 5n + 4, 3n + 4}
and
S = {8(n + 1), 8n + 7, 8n + 6,..., 7n + 9, 7n + 8, 7(n + 1),... , 2(3n + 4), 2n + 3, 2n + 4, ..., 3n + 2, 3(n + 1), 3n + 5, 3n + 6,..., 4n + 3, n + 1}.
Clearly, R and S are independent sets. The number a = 4n + 5 satisfies f (u) < a < f (v) for every ordered pair (u, v) e R x S. Therefore, tf is an a-valuation of Cfn U Cfn (see Cf3 U Cf3 in Fig. 2). □
7 18
Figure 2. An a-valuation of Cf3 u Cf3.
Theorem 4. Let Cfm U Cf1, n e {4m, 4m - 1,4m - 2}, be the disjoint union of hairy cycles. If there is a function
0 : V(Cfm U Cf1) ^ {0,1, 2,... , 2(4m + n)},
then
(i) the graph C4m U C^f1, n e {4m, 4m - 2}, admits an a-valuation;
(ii) the graph Cfm U Cfm_ m > 1, admits a graceful valuation.
Proof. Let Cfl be the graph (a hairy cycle) obtained by adding a pendant vertex to each vertex of the cycle of order 4m. To prove this theorem, we need to prove the following claims.
Claim 1. The graph Cfl U Cfl, m = n and m > 1, admits an a-valuation.
Claim 2. The graph Cfl U Cfl_2 admits an a-valuation.
Claim 3. The graph C4m U C4m —1, m > 1, admits a graceful valuation.
Before proving the claims, we fix a labeling of the hairy cycle C4m in C4m U C^f1. Because, throughout the proof, the labeling of the first part Cm is the same. For each t € {4m, 4m — 1, 4m — 2}, we need to define a labeling 0 on
Cfm U cf1 and prove that this is an a-labeling. So, first, we define the labeling of the first part of the union as follows.
Let u and Vj, i = 1,2,...,4m, be the vertices of the cycle and leaves of C4m, respectively.
Then, 4
0(Uj) =
=
i — 1 if i < 2m and
i if i > 2m and
2(4m +1) — (i — 1) if i is even,
2(4m +1) — (i — 1) if i is odd,
is odd, is odd,
i — 1
if i < 2m and i is even, if i > 2m and i is even.
Next, we define the edge labeling g on the edges of C4m by
for uv € E as follows:
g(uv) = |0(u) — 0(v)|
g(ui Uj+1) =
g(UiVj ) =
2(4m +1) — 2i + 1 if i< 2m, 2(4m +1) — 2i if 2m < i < 4m,
g(u4m, u1) = 4m + 2t + 1,
2(4m +1 — i + 1) if i < 2m,
2(4m + t) — 2(i — 1) — 1 if i > 2m.
Proof of Claim 1. This claim holds for only m = n.
Let xj and y^ i = 1,2,..., 4n, be the vertices of the cycle and leaves of C4n (the second part). Clearly,
i v (Cim u Cfm )i = iE(C4m U C4m )i = 16m.
We now define the labeling of C|m as follows.
Case 1: m is even. We label the vertices by 0(x4m) = 10m,
i is odd,
i < m and i is even, m < i < 2m and i is even, 2m < i < 4m — 2 and i is even,
1m, 0(y4m) = 2m,
i is odd, and i is odd, m — 1, 3m — 1 < i < 4m — 1, and i is odd, and i is even.
It can be verified that all vertices of the graph are labeled and all labels are distinct. Label the set
0(yi) =
4m + i
if
0(Xj) =
12m + 1 — i if 12m — i if
12m — 1 — i
if
12m — (i — 1) 12m — i 12m — 1 — i
4m + i
0(y3m-1) = 1: if i < m and if m < i < 2m if 2m < i < 3i if i < 4m — 2
E of edges in the graph as follows:
{2(4m - i) if i < m,
2(4m - i) - 1 if m < i < 2m - 1,
2(4m - i) - 2 if 2m - 1 < i < 4m - 2,
g(X4m_lX4m) = 2m + 1, g(X4mXi) = 6m - 1,
{2(4m - i) + 1 if i < m, 2(4m - i) if m < i < 2m,
2(4m - i) - 1 if 2m < i < 3m - 1, 3m - 1 < i < 4m - 1. g(X3m_1 y3m-l) = 4m + 1, g(X4m ^4m) = 8m.
Suppose that m = n and m is even. Then, the labeling of C4m U C^ is a graceful valuation. Moreover, the labeling of C^m U C4m is actually an a-valuation with the critical value 2m - 1, and the number 9m/4 is not assigned to any vertex of Cm U C4m.
Case 2: m is odd. If m = 1, the labeling follows from Theorem 2. If m > 2, the labeling is defined as follows:
0(w3m) = 7m + 2, 0(w4m) = 10m, 4m + i if i < m and i is odd,
4m + 1 + i if m < i < 3m, 3m < i < 4m - 1, and i is odd, 12m + 1 - i if i < 2m and i is even, 12m - i if 2m < i < 4m - 2 and i is even,
0(Z3m+1) = 5m + 1, 0(z4m) = 2m,
12m + 1 - i if i < 2m and i is odd,
12m - i if 2m < i < 4m - 1 and i is odd,
4m + i if i < m and i is even,
4m + 1 + i if m < i < 3m, 3m + 1 < i < 4m - 2, and i is even.
=
) =
It can be verified that all the vertices of the graph are labeled and the labels are distinct. We now construct labels for the set E of edges in the graph as follows:
g(W3m-1 W3m) = 2m - 1, g(w3mW3m+l) = 2m - 3, g(w4m_1 W4m) = 2m, g(W4mWi) = 6m - 1,
{2(4m - i) if i < m,
2(4m - i) - 1 if m < i < 2m,
2(4m - i) - 2 if 2m < i < 3m - 1, 3m < i < 4m - 2, g(W3mZ3m) = 2(m - 1), g(W3m+1Z3m+1) = 4m - 2, g(W4mZ4m) = 8m,
{2(4m - i) + 1 if i < m,
2(4m - i) if m < i < 2m,
2(4m - i) - 1 if 2m < i < 3m, 3m + 1 < i < 4m - 1.
Through the close examination of the above function it can be seen that the induced edge labeling is bijective. It is clear that all the vertex labels are distinct. The edge labels are computed from these vertex labels and are also found to be distinct from 1 to 16m. Therefore, Cfm U Cfn, where m is odd and m = n, is a graceful valuation. Moreover, the labeling of C4m U C4m is actually an a-valuation with the critical value 2m, and the number 7m + 4/4 is not assigned to any vertex of Cfm U Cm. This completes the proof of Claim 1.
Proof of Claim 2. Let aj and bj, i = 1,2,..., 4m — 2, be the vertices of the cycle and leaves
of Cfm - 2.Define the labelingof C4fm _2as follows:
0(«2m) = 6m + 1,
0(flj) =
)=
3(4m — 1) — i 4m + i
4m + i
if i is odd, if 1 < i < 2m — 1,
2m < i < 4m — 2, and i is even,
0(b2m+1) = 2m, if 1 < i < 2m — 1, 2m + 1 < i < 4m — 2, and i is odd,
3(4m — 1) — i if i is even.
Moreover, this produces the edge labels of C4m_2:
g(a4m_2 01) = 4m — 2, g(a2m^2m) = 4(m — 1), g(a2m+1^2m+1) = 4m — 2, 2(4m — 2 — i) if i < 2m — 1, 2m < i < 4m — 2, 8m — 5 — 2i if 2m — 1 < i < 2m,
g(ajaj+1) =
g(ajbj) = 8m — 3 — 2i if i < 2m — 1, i > 2m + 1.
Through these combined labelings of the hairy cycles C4m (defined before Claim 1) and C4m_2 bring out the labeling of C4m U C4m_2, and its induced edge labeling is bijective. It is clear that all the vertex labels are distinct. The edge labels are computed from these vertex labels and are also found to be distinct from 1 to 16m — 4. Therefore, Cfm U Cfm_ 2 is a graceful valuation. Moreover, the labeling of Cfl U Cfl_ _2 is actually an a-valuation with critical value 5m — 2, and the number 3m/2 is not a label of Cfl U Cfl_2 (see Cf1 U Cf1 in Fig. 3). This completes the proof of Claim 2.
26
Figure 3. An a-valuation of Cf1 u Cf1.
Proof of Claim 3. Let wj and zj, i = 1,2,..., 4m — 1, be the vertices of the cycles and leaves of C4SI_1. Clearly,
IV(C4SI U C4SI_ 1 )| = |E(C4* U C4SI_ 1 )| = 16m — 2.
Since the labeling of C^ is defined at the beginning of the proof, we only need to specify a labeling
of Cfm _ 1and cfm and we do this as follows.
Case 1 : m is even. Define
0(x4m-i) = 2(5m - 1),
=
) =
12m — 1 — i 2(6m — 1) — i
4m + i if i < 4m — 3 and i is odd,
12m — 1 — i if i < m and i is even,
2(6m — 1) — i if m < i < 2m and i is even,
3(4m — 1) — i if i > 2m and i is even,
0(y3m-i) = 11m — 2
if i < m and if m < i < 2m
0(x4m-i) = 2m,
i is odd, and i is odd,
3(4m — 1) — i if 2m < i < 3m — 1, 3m — 1 <i< 4m — 1, and i is odd, 4m + i if i is even.
It can be verified that all the vertices of the graph are labeled and the labels are distinct. We now construct the set E of edge labels in the graph as follows:
g(X4m_2X4m_1) = 2m - 1, g(X4m_1 X1) = 3(2m - 1),
{2(4m - 1 - i) if i < m,
2(4m - i) - 3 if m < i < 2m - 1,
2(4m - 2 - i) if 2m - 1 < i < 4m - 3,
g(X3m_ 1 y3m_ 1) = 4m - 1, g(X4m_1 y4m_1) = 2(4m - 1),
{2(4m - i) - 1 if i < m,
2(4m - 1 - i) if m < i < 2m - 1,
2(4m - i) - 3 if 2m - 1 < i < 3m - 1, 3m - 1 < i < 4m - 2. Case 2: m is odd and m > 1. Figure 4 shows a graceful valuation for m = 1.
10
2
5
4
7
Figure 4. A graceful valuation of Cfl u Cfl.
If m > 1, then we define
7m + 2, 0(w4m_1) = 10m - 2, i < m and i is odd,
m < i < 3m, 3m < i < 4m - 1, and i is odd, i < 2m and i is even, i > 2m and i is even,
) =
0(W3m) =
4m + i if
4m + 1 + i if
12m — 1 — i if
2(6m — 1) — i if
0(zi) =
0(Z4m-1) = 2m, 0(Z3m+1) = 5m + 1,
12m — 1 — i if i < 2m and i is odd,
2(6m — 1) — i if 2m < i < 4m — 3 and i is odd,
4m + i if i < m and i is even,
4m + i + 1 if m < i < 3m + 1, 3m + 1 < i < 4m — 1,
and
is even.
It can be verified that all the vertices of the graph are labeled and the labels are distinct. We now construct labels for the set E of edges in the graph as follows:
g(w3m_1 W3m) = 2m — 3, g(w3m W3m+1) = 2m — 5, g(W4m_2 W4m_1) = 2(m — 1), g(w4m_1 W1) = 3(2m — 1),
{2(4m — 1 — i) if i < m, 2(4m — i) — 3 if m < i < 2m,
2(4m — 2 — i) if 2m < i < 3m — 1, 3m < i < 4m — 2, g(W3mZ3m) = 2(m — 2), g(W3m+1Z3m+1) = 4(m — 1), g (W4m_ 1 Z4m_ 1) = 2(4m — 1),
{2(4m — i) — 1 if i < m, 2(4m — 1 — i) if m < i < 2m,
2(4m — i) — 3 if 2m < i < 3m, 3m + 1 < i < 4m — 1.
We see that the labels of the edges of C4m U C4m_1 are distinct. Therefore, it can be easily shown that the graph C^ U C4m_1 has graceful valuations (see Cfl U Cfl in Fig. 5). This completes the proof of Claim 3. □
Figure 5. A graceful valuation of Cfl u cf1.
Theorem 5. The graph obtained by the disjoint union of two isomorphic copies of all hairy cycles C* +2 admits an a-valuation with exactly one missing number p = 4(2m +1) and the critical value a = 2(3m + 2).
Proof. Let i = 1,2,3,..., 4m + 2, and let p (fj) and q» (sj) denote the vertices of the cycle and leaves, respectively, in the first and second copies of C4m+2, respectively. Clearly,
iv (cfm+2 u C4m+2)| = iE(c4m+2 u C^I = 8(2m+1).
Define a function £ : V(Cfm+2 U Cfm+2) ^ {0,1,2,..., 8(2m + 1)} as follows. Figure 6 shows
Si
a
an a-valuation for m = 1.
17
Figure 6. An a-valuation of Cf1 u Cf1.
For m > 1, we label the vertices of Cfl+2 U Cfl+2 as follows:
=
=
16m + 9 — i 16m + 8 — i i—1
C(P4m+2) = 12m + 7,
if i < 2m + 1 and i is even, if 2m + 1 < i < 4m and i is even,
if i is odd,
£(%) =
£(q4m+i) = 12m + 6, £(q4m+2) = 4m + 2,
16m + 9 — i if i < 2m + 1 and i is odd, 16m + 8 — i if 2m + 1 < i < 4m — 1 and i i — 1 if i < 4m and i is even,
is odd,
£(ri) = 4m + 1, C(r2m+3) = 6m + 5, £(^+2) = 10m + 3,
4m + 1 + i if 1 < i < 2m + 1, 2m + 3 <i < 4m + 1, 12m + 6 — i ifi < 2m + 2, 2m + 2 < i < 4m + 2, and
and i is odd, i is even,
£(s2m+i) = 10m + 4, £(s2m+3) = 14m + 7, £(s2m+5) = 10m + 5,
£(Si) =
12m + 6 — i if i < 2m + 1 and i is odd,
12m + 8 — i if 2m + 5 < i < 4m + 2 and i is odd,
4m + 1 + i if 1 < i < 2m + 2 and i is even,
4m + 3 + i if 2m + 2 < i < 4m + 2 and i is even.
Clearly, £ is injective. Now, we prove that the induced labeling
I : E(Cfm+2 U C*) ^ {1, 2,... , 8(2m + 1)}
defined as l(xy) = |£(x) - £(y)| for xy € E(Cfm+2 U Cfm+2) is bijective.
The induced edge labeling I has the following values:
l(P4m+2P1) = 12m + 7, l(p4m+1 P4m+2) = 8m + 7, l( ) = f 16m + 9 — 2i if i < 2m + 1, (PiPi+1) = j 16m + 8 — 2i if 2m + 1 < i < 4m,
I (P4m+1 q4m+1) = 8m + 6, l(p4m+2 94m+2) = 8m + 5, 16m + 10 — 2i if i < 2m + 1,
l(piqi) 1 16m + 9 — 2i if 2m + 1 < i < 4m, l(r1 r2) = 8m + 3, l(r2m+2 ^2^+3) = 4m — 2, 1(^+2^) = 4m + 3, l(r2m+1 r2m+2) = 4m + 1, l(r2m+3r2m+4) = 4m — 3, l(riri+1) = 8m + 4 — 2i, for 1 <i< 2m + 1, 2m + 4 < i < 4m + 1, l(r1S1) = 8m + 4, l(r2m+3r2m+3) = 8m + 2, l(r2m+5S2m+5) = 4m — 1, l(r4m+2S4m+2) = 1, l(r2m+1S2m+1) = 4m + 2, l(r2m+2 S2m+2) = 4m,
{8m + 5 — 2i if 1 < i < 2m + 1,
8m — 2i + 3 if 2m + 4 < i < 4m + 2 and i is even, 8m — 2i + 7 if i > 2m + 5 and i is odd.
It is clear that all the vertex and edge labels are distinct. Therefore, the graph Cfm+2 U Cfm+2 is graceful. Next, we prove that the above graceful function £ is an a-valuation with the missing number p = 4(2m + 1) and the critical value a = 2(3m + 2). Since the vertex set V of Cfm+2 UCfm+2 is partitioned into two sets, V = X U Y, we have
X={0,1,2,..., 4m, 4m + 2,4m + 1,4m + 3,4m + 4,..., 6m + 3,6m + 5,6m + 7,6m + 6,... , 8m + 5} and
Y={8(2m + 1), 16m + 7,16m + 6,..., 14m + 8,14m + 6,..., 12m + 6,12m + 7,12m + 5,12m + 4, ..., 10m + 6,10m + 4,..., 14m + 7,10m + 2,10m + 5,..., 8m + 4}.
Clearly, X and Y are independent sets. The number a = 2(3m + 2) satisfies f (x) < a < f (y) for every ordered pair (x,y) € X x Y. Therefore, £ is an a-valuation of Cfm+2 U C4m+2. □
3. Conclusion
This paper discussed graceful and a-valuations of certain disconnected graphs. Finding general characterizations for such graphs is an open problem. We also propose the following problem.
Problem 3.1. The disjoint union of various hairy cycles C^™ U C^g U ••• U C^ admits an a-labeling.
Acknowledgements
The authors wish to thank the referees for their valuable comments.
REFERENCES
1. Abrham J., Kotzig A. Graceful valuations of 2-regular graphs with two components. Discrete Math.., 1996. Vol. 150, No. 1-3. P. 3-15. DOI: 10.1016/0012-365X(95)00171-R
2. Balakrishnan R., Ranganathan K. A Textbook of Graph Theory, 2nd ed. NY: Springer, 2012. 292 p. DOI: 10.1007/978-1-4614-4529-6
3. Barrientos C. Equitable labeling of corona graphs. J. Comb. Math. Comb. Comput., 2002. Vol. 41. P. 139-149.
4. Barrientos C. Graceful graphs with pendant edges. Australas. J. Comb., 2005. Vol. 33. P. 99-107.
5. Barrientos C., Minion S. Constructing graceful graphs with caterpillars. J. Algorithms Comput. 2016. Vol. 48, No. 1. P. 117-125. DOI: 10.22059/jac.2016.7946
6. Eshghi K., Carter M. Construction of a-valuations of special classes of 2-regular graphs Math. Comput. Sci. Eng, 2001. P. 139-154.
7. Frucht R., Harary F. On the corona of two graphs. Aequationes Math., 1970. Vol. 4. P. 322-325. DOI: 10.1007/BF01844162
8. Frucht R. W., Salinas L. C. Graceful numbering of snakes with constraints on the first label. Ars Comin., 1985. Vol. 20B. P. 143-157.
9. Gallian J. A. A dynamic survey of graph labeling. 23-th ed. Electron. J. Combin., 2023. Art. no. #DS6. P. 1-644. DOI: 10.37236/27
10. Graf A. A new graceful labeling for pendant graphs. Aequat. Math., 2014. Vol. 87. P. 135-145. DOI: 10.1007/s00010-012-0184-4
11. Kumar J., Mishra D., Kumar A., Kumar V. Alpha labeling of cyclic graphs. Int. J. Appl. Comput., 2021. Vol. 7, Art. no. 151. P. 1-7. DOI: 10.1007/s40819-021-01084-5
12. Kumar A, Mishra D, Verma A., Srivastava V. K. Alpha labeling of cyclic graphs. I. ARS Combinatorics, 2021. Vol. 154. P. 257-263.
13. Lakshmi D. R., Vangipuram S. An a-valuation of quadratic graph Q(4, 4k). Proc. Nat. Acad. Sci., India, Sect. A, 1987. Vol. 57, No. 4. P. 576-580.
14. Minion S., Barrientos C. Three Graceful Operations. J. Alg. Comp, 2014. Vol. 45, No. 1. P. 13-24. DOI: 10.22059/jac.2014.7917.
15. Pradhan P., Kumar A. Graceful hairy cycles with pendent edges and some properties of cycles and cycle related graphs. Bull. Calcutta Math.. Soc., 2012. Vol. 104. P. 61-76.
16. Pradhan P., Kumar K. On graceful labeling of some graphs with pendant edges. Gen. Math. Notes, 2014. Vol. 23, No. 2. P. 51-62.
17. Pradhan P., Kumar K. On k-graceful labeling of some graphs. J. Appl. Math. Inform., 2016. Vol. 34, No. 1-2. P. 9-17. DOI: 10.14317/jami.2016.009.
18. Pradhan P., Kumar A., Mishra D. On gracefulness of graphs obtained from hairy cycles. J. Combin. Inform. Syst. Sci., 2010. Vol. 35, P. 471-480.
19. Ringel G. Problem 25. In: Theory of Graphs and its Application. Proc. Symposium Smolenice 1963. Prague, 1964.
20. Ropp D. Graceful labelings of cycles and prisms with pendant points. Congr. Number, 1990. Vol. 75. P. 218-234.
21. Rosa A. On certain valuations of the vertices of a graph. In: Theory of Graphs: Int. Symposium, Rome, July 1966). N.Y. and Dunod Paris: Gordon and Breach, 1967. P. 349-355.
22. Truszczynski M. Graceful unicyclic graphs. Demonstratio Math., 1984. Vol. 17. P. 377-387.
