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On the strength of some trees
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On the strength of some trees Rikio Ichishimaa , Francesc A. Muntaner-Batleb , Akito Oshimab ,∗ a
Department of Sport and Physical Education, Faculty of Physical Education, Kokushikan University, 7-3-1 Nagayama, Tama-shi, Tokyo 206-8515, Japan b Graph Theory and Applications Research Group, School of Electrical Engineering and Computer Science, Faculty of Engineering and Built Environment, The University of Newcastle, NSW 2308, Australia Received 25 March 2019; accepted 3 June 2019 Available online xxxx
Abstract Let G be a graph of order p. A numbering f of G is a labeling that assigns distinct elements of the set {1, 2, . . . , p} to the vertices of G, where each edge uv of G is labeled f (u) + f (v). The strength str f (G) of a numbering f : V (G) → {1, 2, . . . , p} of G is defined by str f (G) = max { f (u) + f (v) |uv ∈ E (G) } , that is, str f (G) is the maximum edge label of G, and the strength str(G) of a graph G itself is { } str (G) = min str f (G) | f is a numbering of G . ( ) The strengths str (T ) and str Tn,k are determined for caterpillars T and k-level complete n-ary trees Tn,k . The strength str (G) is also given for graphs G obtained by taking the corona of certain graphs and an arbitrary number of isolated vertices. The work of this paper suggests an open problem on the strength of trees. c 2019 Kalasalingam University. Production and Hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND ⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Strength; Caterpillar; Tree; Graph labeling; Corona
1. Introduction In this paper, we will consider only finite graphs without loops or multiple edges. We refer the reader to the book by Chartrand and Lesniak [1] for graph-theoretical notation and terminology not described in this paper. The graph with n vertices and no edges is referred to as the empty graph of order n and is denoted by n K 1 . The degree of a vertex v in a graph is number of edges of G incident with v and is denoted by degG v. A vertex of degree 0 is called an isolated vertex and a vertex of degree 1 is an end-vertex of G. The minimum degree of G is the minimum degree among the vertices of G and is denoted by δ (G). Peer review under responsibility of Kalasalingam University. Corresponding author. E-mail addresses: [email protected] (R. Ichishima), [email protected] (F.A. Muntaner-Batle), [email protected] (A. Oshima). ∗
https://doi.org/10.1016/j.akcej.2019.06.002 c 2019 Kalasalingam University. Production and Hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND 0972-8600/⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article as: R. Ichishima, F.A. Muntaner-Batle and A. Oshima, On the strength of some trees, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.06.002.
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For the sake of notational convenience, we will denote the interval of integers k such that i ≤ k ≤ j by simply writing [i, j]. For a graph G of order p and size q, a bijective function f : V (G) ∪ E(G) → [1, p + q] is called an edge-magic labeling if f (u) + f (v) + f (uv) is a constant c ( f ) (called the magic constant) for each uv ∈ E(G). If such a labeling exists, then G is called an edge-magic graph. The notion of edge-magic labelings was first introduced in 1970 by Kotzig and Rosa [2]. These labelings were originally called “magic valuations” by them. These were rediscovered in 1996 by Ringel and Llad´o [3] who coined one of the now popular terms for them: edge-magic labelings. Afterwards, Enomoto et al. [4] defined a slightly restricted version of an edge-magic labeling f of a graph G by requiring that f (V (G)) = [1, |V (G)|]. Such a labeling was called by them super edge-magic . Thus, a super edge-magic graph is a graph that admits a super edge-magic labeling. It is worth to mention that Acharya and Hegde [5] had already discovered such graphs in 1991 under the name of “strongly indexable graphs”. However, they arrived at this concept from a different point view. Their motivation is much clear from the following alternative definition found in [6]. Lemma 1. A graph G of order p and size q is super edge-magic if and only if there exists a bijective function f : V (G) → [1, p] such that the set S = { f (u) + f (v) |uv ∈ E (G) } consists of q consecutive integers. In such a case, f extends to a super edge-magic labeling of G with magic constant k = p + q + s, where s = min (S) and S = [k − ( p + q) , k − ( p + 1)] . The concept of super magic strength was introduced by Avadayappan et al. [7]. The super magic strength sm (G) of a graph G is defined as the minimum of all magic constants c ( f ), where the minimum is taken over all super edge-magic labelings f of G, that is, sm (G) = min {c ( f ) | f is a super edge-magic labeling of G } . It is an immediate consequence of the definition that if G is not a super edge-magic graph or an empty graph, then sm (G) is undefined (or we could define sm (G) = +∞). It is also true that G is a super edge-magic graph if and only if sm (G) < +∞. As the concept of super magic strength is effectively defined only for super edge-magic graphs, this concept was generalized in [8] for any nonempty graph as follows. A numbering f of a graph G of order p is a labeling that assigns distinct elements of the set [1, p] to the vertices of G, where each edge uv of G is labeled f (u) + f (v). The strength str f (G) of a numbering f : V (G) → [1, p] of G is defined by str f (G) = max { f (u) + f (v) |uv ∈ E (G) } , that is, str f (G) is the maximum edge label of G, and the strength str(G) of a graph G itself is { } str (G) = min str f (G) | f is a numbering of G . A numbering f of a graph G for which str f (G) = str (G) is called a strength labeling of G. If G is an empty graph, then str (G) is undefined (or we could define str (G) = +∞). There are several sharp bounds for the strength of a graph in terms of well-known invariants in graph theory (see [8]). Among others, the following result that provides a lower bound for str (G) in terms of the order and the minimum degree is particularly useful. Lemma 2. For every graph G of order p with δ (G) ≥ 1, str (G) ≥ p + δ (G) . The following results were obtained in [8] for the paths Pn and stars K 1,n−1 of order n. These classes of graphs illustrate the sharpness of the bound given in Lemma 2. Theorem 1. For every integer n ≥ 2, str (Pn ) = n + 1. Please cite this article as: R. Ichishima, F.A. Muntaner-Batle and A. Oshima, On the strength of some trees, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.06.002.
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Theorem 2. For every integer n ≥ 2, ( ) str K 1,n−1 = n + 1. We close this section with suggestions for further reading. A survey article on graph labelings and related topics can be found in Gallian [9], and a book by Baˇca and Miller [10] is another useful resource. Readers interested in more information on super edge-magic graphs should consult the books by L´opez and Muntaner-Batle [11] as well as Marr and Wallis [12], which also include information on other kinds of graph labelings. 2. The strength of Caterpillars In this section, we investigate the strength of caterpillars, beginning with double stars. The double star Sm,n is a tree obtained by joining the centers of two disjoint stars K 1,m and K 1,n with an edge. Theorem 3. For every two positive integers m and n, ( ) str Sm,n = m + n + 3. ( ) Proof. The inequality str Sm,n ≥ m + n + 3 is a direct consequence of Lemma ( ) 2. In order to verify the inequality in the other direction, it suffices to find a numbering f of Sm,n with str f Sm,n = m + n + 3. Assume, without loss of generality, that 1 ≤ m ≤ n, and define the double star Sm,n with ( ) V Sm,n = {x, y} ∪ {xi |i ∈ [1, m] } ∪ {yi |i ∈ [1, n] } and ( ) E Sm,n = {x y} ∪ {x xi |i ∈ [1, m] } ∪ {yyi |i ∈ [1, n] } . ( ) Now, consider the labeling f : V Sm,n → [1, m + n + 2] such that f (x) = 1, f (y) = 2, { i + n + 2 if v = xi and i ∈ [1, m] , f (v) = i +2 if v = yi and i ∈ [1, n] . Notice that f assigns distinct elements of the set [1, m + n + 2] to the vertices of Sm,n . Also, notice that { f (x) + f (y)} = {3} , { f (x) + f (xi ) |i ∈ [1, m] } = [n + 4, m + n + 3] , { f (y) + f (yi ) |i ∈ [1, n] } = [5, n + 4] . Thus, f has the property that ⏐ ( ) { ( )} str f Sm,n = max f (u) + f (v) ⏐uv ∈ E Sm,n = f (x) + f (xm ) = m + n + 3. ( ) Consequently, str Sm,n = m + n + 3. □ A caterpillar is a tree T with the property that the removal of the end-vertices of T results in a path. This path is referred to as the spine of the caterpillar. If the spine is trivial, the caterpillar is a star; if the spine is K 2 , then the caterpillar is a double star. The next result includes caterpillars whose spine is both trivial and K 2 . Realize that the way of defining the spine provided in the proof is intended to make the vertex labeling easy to describe. Theorem 4. For every caterpillar T of order p, str (T ) = p + 1. Proof. In light of Theorems 2 and 3, it suffices to prove the theorem for caterpillars whose spine contains at least 3 vertices and both end-vertices of the spine have end-vertices attached. The lower bound for str (T ) is obtained immediately from Lemma 2. Please cite this article as: R. Ichishima, F.A. Muntaner-Batle and A. Oshima, On the strength of some trees, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.06.002.
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To establish the upper bound for str (T ), let T be the caterpillar with (m ) } ⋃ { j⏐ V (T ) = {u i |i ∈ [1, m] } ∪ vi ⏐ j ∈ [1, n i ] i=1
and (m ) } ⋃{ j ⏐⏐ E (T ) = {u i u i+1 |i ∈ [1, m − 1] } ∪ u i vi j ∈ [1, n i ] . i=1
Then T has the spine S with V (S) = {u i |i ∈ [1, m] } and E (S) = {u i u i+1 |i ∈ [1, m − 1] } . Note that if we define a permutation π of V (S) by π (u i ) = x j (i ∈ [1, m] and j ∈ [1, m]) , where n ′j = degT u i − deg S u i = n i with n ′1 ≥ n ′2 ≥ · · · ≥ n ′m , then π is an involutive automorphism of S, and we have V (S) = {xi |i ∈ [1, m] }. Hence, every edge of S joins vertices xs and xt with j = i + 1 and i ∈ [1, m − 1] whenever π −1 (xs ) = u j and π −1 (xt ) = u i or π −1 (xs ) = u i and π −1 (xt ) = u j . In the case that n s = n t with 1 ≤ s < t ≤ m, we may assume that π (u s ) is located to the left of π (u t ) in the drawing of T in the plane. The preceding construction allows us to define the caterpillar T with (m ) } ⋃ { j⏐ [ ] V (T ) = {xi |i ∈ [1, m] } ∪ yi ⏐ j ∈ 1, n i′ i=1
and (m ) ⋃ { j⏐ ⏐ −1 } [ ]} −1 ′ xi yi ⏐ j ∈ 1, n i . E (T ) = xs xt ⏐π (xs ) π (xt ) ∈ E (S) ∪ {
i=1
[ ] If we let σi = k=1 n ′k , then p = m + σm . Notice that if n i′ = 0 for some i ∈ [1, m], then we treat 1, n i′ as the empty set. Thus, assume that n i′ ̸= 0 for some i ∈ [1, m]; otherwise, S ∼ = Pm and the result follows from Theorem 1. It remains to show the existence of a numbering f of T for which str f (T ) = p + 1. To complete this, there are two cases to proceed according to the possible values for the integer m. ∑i
Case 1: Assume that m ≤ ⌊ p/2⌋, and consider the labeling f : V (T ) → [1, p] such that { i if v = xi and i ∈ [1, m] , [ ] f (v) = j p − σi + j if v = yi , i ∈ [1, m] and j ∈ 1, n i′ . Then f assigns distinct elements of the set [1, p] to the vertices of T . In particular, f assigns distinct elements of the set [1, m] to the vertices of S. This implies that max { f (u) + f (v) |uv ∈ E (S) } ≤ m + (m − 1) = 2m − 1 ≤ 2 ⌊ p/2⌋ − 1 < p. However, max { f (u) + f (v) |uv ∈ E (T ) \E (S) } = max E i j , Please cite this article as: R. Ichishima, F.A. Muntaner-Batle and A. Oshima, On the strength of some trees, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.06.002.
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⏐ ]} [ where E i j = p − σi + i + j ⏐i ∈ [1, m] and j ∈ 1, n i′ . At this point, let i be the integer so that n i′ ̸= 0 with i ∈ [1, m]. Then n ′1 ≥ n ′2 ≥ · · · ≥ n i′ ≥ 1, which implies that σi−1 ≥ i − 1. Thus, {
max { f (u) + f (v) |uv ∈ E (T ) \E (S) } ≤ max E i j ≤ p − σi + i + n i′ = p − σi−1 + i ≤ p − (i − 1) + i = p + 1. Consequently, max { f (u) + f (v) |uv ∈ E (T ) } ≤ p + 1, so f has the property that str f (T ) = p + 1. ⏐{ ⏐ }⏐ Case 2: Assume that m ≥ ⌊ p/2⌋ + 1, and let l = ⏐ n i′ ⏐i ∈ [1, m] and n i′ = 0 ⏐. Then n i′ ≥ 1 for i ∈ [1, m − l] and n i′ = 0 for i ∈ [m − l + 1, m]. With this knowledge in hand, consider the labeling f : V (T ) → [1, p] such that ⎧ if v = xi and i ∈ [1, m − l] , ⎪ ⎪ i ⎨ m + 1 − i if v = xm−l−1+2i and i ∈ [1, ⌈l/2⌉] , f (v) = m −l +i if v = xm−l+2i and i ∈ [1, ⌊l/2⌋] , ⎪ ⎪ [ ] ⎩ j p − σi + j if v = yi , i ∈ [1, m − l] and j ∈ 1, n i′ . Then f assigns distinct elements of the set [1, p] to the vertices of T . Thereby, f assigns distinct elements of the set [1, m] to the vertices of S as follows: { f (xi ) |i ∈ [1, m − l] } = [1, m − l] , { f (xm−l−1+2i ) |i ∈ [1, ⌈l/2⌉] } = [m − ⌈l/2⌉ + 1, m] , { f (xm−l+2i ) |i ∈ [1, ⌊l/2⌋] } = [m − l + 1, m − ⌈l/2⌉] . For the rest of the proof, we will examine the induced edge labels given by f (u) + f (v) for each uv ∈ E (T ). Then, in a similar manner to Case 1, we have max { f (u) + f (v) |uv ∈ E (T ) \E (S) } ≤ max E i j ≤ p − σi + i + n i′ = p − σi−1 + i ≤ p − (i − 1) + i = p + 1, ⏐ [ ]} where E i j = p − σi + i + j ⏐i ∈ [1, m − l] and j ∈ 1, n i′ . However, there are three subcases to pursue according to the possibility for the edge xs xt ∈ E (S). ⏐ { } Subcase 2.1: Let W1 = xs xt ∈ E (S) ⏐n ′ ̸= 0 and n ′ ̸= 0 . Then {
s
t
max { f (u) + f (v) |uv ∈ W1 } = f (xm−l−1 ) + f (xm−l ) = (m − l − 1) + (m − l) = 2m − 2l − 1. However, since T has ( p − m) end-vertices and (m − l) vertices of S have end-vertices, it follows that p−m ≥ m−l, that is, l ≥ 2m − p.
(1)
This implies that 2m − 2l − 1 ≤ 2 p − 2m − 1.
(2)
Since m ≥ ⌊ p/2⌋ + 1, it also follows from (2) that 2 p − 2m − 1 ≤ 2 p − 2 ⌊ p/2⌋ − 3.
(3)
Please cite this article as: R. Ichishima, F.A. Muntaner-Batle and A. Oshima, On the strength of some trees, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.06.002.
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Thus, the fact that 2 ⌊ p/2⌋ ≥ p − 1 together with (3) implies that max { f (u) + f (v) |uv ∈ W1 } ≤ 2 p − 2 ⌊ p/2⌋ − 3 ≤ p − 2. ⏐ { } Subcase 2.2: Let W2 = xs xt ∈ E (S) ⏐n ′s ̸= 0 and n ′t = 0 . Then max { f (u) + f (v) |uv ∈ W2 } = f (vm−l ) + f (vm−l+1 ) = (m − l) + m = 2m − l. Thus, it follows from (1) that max { f (u) + f (v) |uv ∈ W2 } ≤ 2m − l ≤ 2m − (2m − p) = p. ⏐ } { Subcase 2.3: Let W3 = xs xt ∈ E (S) ⏐n ′s = 0 and n ′t = 0 . Then max { f (u) + f (v) |uv ∈ W3 } = f (xm−l−1+2i ) + f (xm−l+2i ) = (m − l + i) + (m + 1 − i) = 2m − l + 1 for each i ∈ [1, ⌊l/2⌋]. Thus, it follows from (1) that max { f (u) + f (v) |uv ∈ W3 } ≤ 2m − l + 1 ≤ 2m − (2m − p) + 1 = p + 1. Consequently, max { f (u) + f (v) |uv ∈ E (T ) } ≤ p + 1, so f has the property that str f (T ) = p + 1.
□
3. The strength of complete n-ary trees The k-level complete n-ary tree Tn,k is a tree in which the ith level consists of n i−1 vertices and each vertex in level i < k has n ‘sons’ at level i + 1. If n = 2, then Tn,k is referred to as a k-level complete binary tree. The strength of T2,k depends on k as the next result indicates. The proof involves the concept of distance in a graph. For a connected graph G and pair u, v ∈ V (G), the distance d (u, v) between u and v is the length of a shortest u − v path in G. Theorem 5. For every integer k ≥ 2, ( ) str T2,k = 2k . ⏐ ( ( ) )⏐ k ⏐ ⏐ Proof. ( ) The lower bound for str T2,k follows immediately from Lemma 2, since V T2,k = 2 − 1 and δ T2,k = 1. ( ) To the upper bound for str T2,k , it suffices to show the existence of a numbering f of T2,k for which ( establish ) str f T2,k = 2k . Let T2,k be given as a rooted tree in the plane. Then T2,k has a unique vertex of degree 2, so chose this distinct vertex to be the root of T2,k and denoted it by r . Of course, if T2,k is drawn in the plane, then by letting { ( ) } Si = v ∈ V T2,k |i = d (r, v) + 1 , the vertices in each level i ≥ 2 can be ordered from left to right using the integers 1, 2, . . . , |Si |. Since the ith level consists of 2i vertices, it follows that |Si | = 2i−1 . Thus, Si can be written as { ⏐ [ ]} j Si = vi ⏐ j ∈ 1, 2i−1 , Please cite this article as: R. Ichishima, F.A. Muntaner-Batle and A. Oshima, On the strength of some trees, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.06.002.
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where i represents the level that the vertex occupies in T2,k and j represents the place that the vertex in level i occupies in the level starting to count from left to right. If we let { ( ) } E i = uv ∈ E T2,k |u ∈ Si and v ∈ Si+1 , then |E i | = 2i and the edges of E i can also be ordered from left to right in ascending order starting with 1 and j ending up with 2i for each set E i . Thus, we will write ei for the edge in the set E i that occupies position j when j j counting the edges of E i from left to right. This implies that if ei , eil ∈ E i with j < l, then ei is located to the left of eil in the drawing of T2,k in the plane. ( ) ( ) 4 )that str T2,3 =] Since T2,2 ∼ = P3 , it follows from Theorem 1 that str T2,2 = 22 . It also follows from Theorem ( [ 23 , since T2,3 is a caterpillar. Let k be an integer with k ≥ 4, and consider the labeling f : V T2,k → 1, 2k − 1 such that ⎧ [ ] k−1 − 2i + j if i ∈ [1, k − 2] and j ∈ 1, 2i−1 , ⎨ 2 ( ) ⎪ [ ] j 2k−2 + 1 − j if i = k − 1 and j ∈ 1, 2k−2 , f vi = [ ] ⎪ ⎩ 2k−1 − 1 + j if i = k and j ∈ 1, 2k−1 . This implies that [ ] { f (v) |v ∈ Sk−1 } = 1, 2k−2 , [ ] { f (v) |v ∈ Si and i ∈ [1, k − 2] } = 2k−2 + 1, 2k−1 − 1 , [ ] { f (v) |v ∈ Sk } = 2k−1 , 2k − 1 . [ k ] ∑k |Si | = 2k − 1, it follows that f assigns distinct elements of the Since i=1 ( set ) 1, 2 − 1 to the vertices of T2,k . ( ) j j For each ei = uv ∈ E T2,k , where u ∈ Si and v ∈ Si+1 , we write f ei for the induced edge label given by ( ) [ ] ( ) j f (u) + f (v). Notice then that for each j, l ∈ 1, 2i−1 with j < l, we have f ei ≤ f eil if i ∈ [1, k − 1] with ( ) ( ) j i ̸= k − 2, whereas we have f ei ≥ f eil if i = k − 2. Thus, f has the property that { ( )⏐ [ ]} j ⏐ max f ei ⏐i ∈ [1, k − 2] and j ∈ 1, 2i−1 { ( ) ( ) ( 1 ) ( 1 )} = max f v11 + f v22 , f vk−2 + f vk−1 ( ) ( ) = f v11 + f v22 ( ) ( ) = 2k−1 − 1 + 2k−1 − 2 = 2k − 3 and { ( )⏐ ( k−1 ) ( k−2 ) [ ]} ⏐ j 2 max f ek−1 ⏐ j ∈ 1, 2i−1 = f vk−1 + f vk2 ( ) = 1 + 2k−1 − 1 + 2k−1 = 2k , implying that ⏐ ( ) { ( )} str f T2,k = max f (u) + f (v) ⏐uv ∈ E T2,k = 2k . ( ) Consequently, str T2,k = 2k . □ The strength of Tn,k is also determined by considering a drawing of Tn,k as a rooted tree in the plane; the proof is quite similar to that of Theorem 5 and will not be included. Theorem 6. For every two integers n ≥ 2 and k ≥ 2, ( ) nk − 1 + 1. str Tn,k = n−1 Please cite this article as: R. Ichishima, F.A. Muntaner-Batle and A. Oshima, On the strength of some trees, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.06.002.
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4. The strength of the corona of graphs The corona of two graphs was introduced by Frucht and Harary [13]. The corona G ⊙ H of two graphs G and H is defined as the graph obtained by taking one copy of G (which has order p) and p copies of H , and then joining the ith vertex of G to every vertex in the ith copy of H . It is now possible to present the following result. Theorem 7. Let G be a graph of order p with δ (G) ≥ 1. If str (G) = p + δ (G), then str (G ⊙ n K 1 ) = (n + 1) p + 1 for every positive integer n. Proof. Suppose that G is a graph of order p with δ (G) ≥ 1, and let f be a strength labeling of G with str (G) = p + δ (G). Assume that f has the property that f (vi ) = i for each i ∈ [1, p], where V (G) = {vi |i ∈ [1, p] }. Further, let H ∼ = G ⊙ n K 1 , and define the graph H with } { ⏐ j V (H ) = V (G) ∪ wi ⏐i ∈ [1, p] and j ∈ [1, n] and { } j⏐ E (H ) = E (G) ∪ vi wi ⏐i ∈ [1, p] and j ∈ [1, n] . Then |V (H )| = (n + 1) p and δ (H ) = 1. Thus, the lower bound for str (H ) follows from Lemma 2. To establish the upper bound for str (H ), consider the labeling g : V (H ) → [1, (n + 1) p] such that { i if x = vi and i ∈ [1, p] , g (x) = j (n + 1) p − ni + j if x = wi , i ∈ [1, p] and j ∈ [1, n] . Then {g (vi ) |i ∈ [1, p] } = [1, p] and } { ( )⏐ j ⏐ g wi ⏐i ∈ [1, p] and j ∈ [1, n] = [ p + 1, (n + 1) p] . This implies that g assigns distinct elements of the set [1, (n + 1) p] to the vertices of H . It is now immediate that p + δ (G) ≤ p + ( p − 1) = 2 p − 1 < (n + 1) p + 1 for every positive integer n. Moreover, g has the property that strg (H ) = max {g (u) + g (v) |uv ∈ E (H ) } ( ) = g w1n + g (v1 ) = (n + 1) p + 1. Consequently, str (H ) = (n + 1) p + 1.
□
The preceding result is of particular interest, since there are infinitely many graphs G for which str (G) = |V (G)| + δ (G) (see [8] for a detailed list of graphs). Applying it with such classes of graphs repeatedly, we obtain other classes of graphs H for which str (H ) = |V (H )| + 1 again. 5. Conclusions In this paper, we have shown that every caterpillar and k-level complete n-ary tree attains the bound provided in Lemma 2. We have also established a formula for the strength of the corona of certain graphs and an arbitrary number of isolated vertices (see Theorem 7). A new tree can be created by taking the corona with isolated vertices. Therefore, applying Theorem 7 with the classes of trees studied in this paper, we obtain other classes of trees T for which str (T ) = |V (T )| + 1. These trees include a class of lobsters (a lobster is a tree T with the property that the removal of the end-vertices of T results in a caterpillar). This motivates us to propose the following problem. Problem 1. For every lobster T , determine the exact value of str (T ). Please cite this article as: R. Ichishima, F.A. Muntaner-Batle and A. Oshima, On the strength of some trees, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.06.002.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
G. Chartrand, L. Lesniak, Graphs & digraphs, in: Wadsworth & Brook/Cole Advanced Books and Software, Monterey, Calif., 1986. A. Kotzig, A. Rosa, Magic valuations of finite graphs, Canad. Math. Bull. 13 (1970) 451–461. G. Ringel, A. Lladó, Another tree conjecture, Bull. Inst. Combin. Appl. 18 (1996) 83–85. H. Enomoto, A. Lladó, T. Nakamigawa, G. Ringel, Super edge-magic graphs, SUT J. Math. 34 (1998) 105–109. B.D. Acharya, S.M. Hegde, Strongly indexable graphs, Discrete Math. 93 (1991) 123–129. R.M. Figueroa-Centeno, R. Ichishima, F.A. Muntaner-Batle, The place of super edge-magic labelings among other classes of labelings, Discrete Math. 231 (2001) 153–168. S. Avadayappan, P. Jeyanthi, R. Vasuki, Super magic strength of a graph, Indian J. Pure Appl. Math. 32 (2001) 1621–1630. R. Ichishima, F.A. Muntaner-Batle, A. Oshima, Bounds for the strength of graphs, Australas. J. Combin. 72 (3) (2018) 492–508. J.A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. (2018) #DS6. M. Baˇca, M. Miller, Super Edge-Antimagic Graphs: A Wealth of Problems and Some Solutions, Brown Walker Press, Boca Raton, FL, USA, 2007. S.C. López, F.A. Muntaner-Batle, Graceful, Harmonious and Magic Type Labelings: Relations and Techniques, Springer, New York, 2016. A.M. Marr, W.D. Wallis, Magic Graphs, second ed., Birkhäuser/Springer, New York, 2013. R. Frucht, F. Harary, On the corona of two graphs, Aequationes Math. 4 (1970) 322–325.
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On the strength of some trees Rikio Ichishimaa , Francesc A. Muntaner-Batleb , Akito Oshimab ,∗ a
Department of Sport and Physical Education, Faculty of Physical Education, Kokushikan University, 7-3-1 Nagayama, Tama-shi, Tokyo 206-8515, Japan b Graph Theory and Applications Research Group, School of Electrical Engineering and Computer Science, Faculty of Engineering and Built Environment, The University of Newcastle, NSW 2308, Australia Received 25 March 2019; accepted 3 June 2019 Available online xxxx
Abstract Let G be a graph of order p. A numbering f of G is a labeling that assigns distinct elements of the set {1, 2, . . . , p} to the vertices of G, where each edge uv of G is labeled f (u) + f (v). The strength str f (G) of a numbering f : V (G) → {1, 2, . . . , p} of G is defined by str f (G) = max { f (u) + f (v) |uv ∈ E (G) } , that is, str f (G) is the maximum edge label of G, and the strength str(G) of a graph G itself is { } str (G) = min str f (G) | f is a numbering of G . ( ) The strengths str (T ) and str Tn,k are determined for caterpillars T and k-level complete n-ary trees Tn,k . The strength str (G) is also given for graphs G obtained by taking the corona of certain graphs and an arbitrary number of isolated vertices. The work of this paper suggests an open problem on the strength of trees. c 2019 Kalasalingam University. Production and Hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND ⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Strength; Caterpillar; Tree; Graph labeling; Corona
1. Introduction In this paper, we will consider only finite graphs without loops or multiple edges. We refer the reader to the book by Chartrand and Lesniak [1] for graph-theoretical notation and terminology not described in this paper. The graph with n vertices and no edges is referred to as the empty graph of order n and is denoted by n K 1 . The degree of a vertex v in a graph is number of edges of G incident with v and is denoted by degG v. A vertex of degree 0 is called an isolated vertex and a vertex of degree 1 is an end-vertex of G. The minimum degree of G is the minimum degree among the vertices of G and is denoted by δ (G). Peer review under responsibility of Kalasalingam University. Corresponding author. E-mail addresses: [email protected] (R. Ichishima), [email protected] (F.A. Muntaner-Batle), [email protected] (A. Oshima). ∗
https://doi.org/10.1016/j.akcej.2019.06.002 c 2019 Kalasalingam University. Production and Hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND 0972-8600/⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article as: R. Ichishima, F.A. Muntaner-Batle and A. Oshima, On the strength of some trees, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.06.002.
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For the sake of notational convenience, we will denote the interval of integers k such that i ≤ k ≤ j by simply writing [i, j]. For a graph G of order p and size q, a bijective function f : V (G) ∪ E(G) → [1, p + q] is called an edge-magic labeling if f (u) + f (v) + f (uv) is a constant c ( f ) (called the magic constant) for each uv ∈ E(G). If such a labeling exists, then G is called an edge-magic graph. The notion of edge-magic labelings was first introduced in 1970 by Kotzig and Rosa [2]. These labelings were originally called “magic valuations” by them. These were rediscovered in 1996 by Ringel and Llad´o [3] who coined one of the now popular terms for them: edge-magic labelings. Afterwards, Enomoto et al. [4] defined a slightly restricted version of an edge-magic labeling f of a graph G by requiring that f (V (G)) = [1, |V (G)|]. Such a labeling was called by them super edge-magic . Thus, a super edge-magic graph is a graph that admits a super edge-magic labeling. It is worth to mention that Acharya and Hegde [5] had already discovered such graphs in 1991 under the name of “strongly indexable graphs”. However, they arrived at this concept from a different point view. Their motivation is much clear from the following alternative definition found in [6]. Lemma 1. A graph G of order p and size q is super edge-magic if and only if there exists a bijective function f : V (G) → [1, p] such that the set S = { f (u) + f (v) |uv ∈ E (G) } consists of q consecutive integers. In such a case, f extends to a super edge-magic labeling of G with magic constant k = p + q + s, where s = min (S) and S = [k − ( p + q) , k − ( p + 1)] . The concept of super magic strength was introduced by Avadayappan et al. [7]. The super magic strength sm (G) of a graph G is defined as the minimum of all magic constants c ( f ), where the minimum is taken over all super edge-magic labelings f of G, that is, sm (G) = min {c ( f ) | f is a super edge-magic labeling of G } . It is an immediate consequence of the definition that if G is not a super edge-magic graph or an empty graph, then sm (G) is undefined (or we could define sm (G) = +∞). It is also true that G is a super edge-magic graph if and only if sm (G) < +∞. As the concept of super magic strength is effectively defined only for super edge-magic graphs, this concept was generalized in [8] for any nonempty graph as follows. A numbering f of a graph G of order p is a labeling that assigns distinct elements of the set [1, p] to the vertices of G, where each edge uv of G is labeled f (u) + f (v). The strength str f (G) of a numbering f : V (G) → [1, p] of G is defined by str f (G) = max { f (u) + f (v) |uv ∈ E (G) } , that is, str f (G) is the maximum edge label of G, and the strength str(G) of a graph G itself is { } str (G) = min str f (G) | f is a numbering of G . A numbering f of a graph G for which str f (G) = str (G) is called a strength labeling of G. If G is an empty graph, then str (G) is undefined (or we could define str (G) = +∞). There are several sharp bounds for the strength of a graph in terms of well-known invariants in graph theory (see [8]). Among others, the following result that provides a lower bound for str (G) in terms of the order and the minimum degree is particularly useful. Lemma 2. For every graph G of order p with δ (G) ≥ 1, str (G) ≥ p + δ (G) . The following results were obtained in [8] for the paths Pn and stars K 1,n−1 of order n. These classes of graphs illustrate the sharpness of the bound given in Lemma 2. Theorem 1. For every integer n ≥ 2, str (Pn ) = n + 1. Please cite this article as: R. Ichishima, F.A. Muntaner-Batle and A. Oshima, On the strength of some trees, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.06.002.
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Theorem 2. For every integer n ≥ 2, ( ) str K 1,n−1 = n + 1. We close this section with suggestions for further reading. A survey article on graph labelings and related topics can be found in Gallian [9], and a book by Baˇca and Miller [10] is another useful resource. Readers interested in more information on super edge-magic graphs should consult the books by L´opez and Muntaner-Batle [11] as well as Marr and Wallis [12], which also include information on other kinds of graph labelings. 2. The strength of Caterpillars In this section, we investigate the strength of caterpillars, beginning with double stars. The double star Sm,n is a tree obtained by joining the centers of two disjoint stars K 1,m and K 1,n with an edge. Theorem 3. For every two positive integers m and n, ( ) str Sm,n = m + n + 3. ( ) Proof. The inequality str Sm,n ≥ m + n + 3 is a direct consequence of Lemma ( ) 2. In order to verify the inequality in the other direction, it suffices to find a numbering f of Sm,n with str f Sm,n = m + n + 3. Assume, without loss of generality, that 1 ≤ m ≤ n, and define the double star Sm,n with ( ) V Sm,n = {x, y} ∪ {xi |i ∈ [1, m] } ∪ {yi |i ∈ [1, n] } and ( ) E Sm,n = {x y} ∪ {x xi |i ∈ [1, m] } ∪ {yyi |i ∈ [1, n] } . ( ) Now, consider the labeling f : V Sm,n → [1, m + n + 2] such that f (x) = 1, f (y) = 2, { i + n + 2 if v = xi and i ∈ [1, m] , f (v) = i +2 if v = yi and i ∈ [1, n] . Notice that f assigns distinct elements of the set [1, m + n + 2] to the vertices of Sm,n . Also, notice that { f (x) + f (y)} = {3} , { f (x) + f (xi ) |i ∈ [1, m] } = [n + 4, m + n + 3] , { f (y) + f (yi ) |i ∈ [1, n] } = [5, n + 4] . Thus, f has the property that ⏐ ( ) { ( )} str f Sm,n = max f (u) + f (v) ⏐uv ∈ E Sm,n = f (x) + f (xm ) = m + n + 3. ( ) Consequently, str Sm,n = m + n + 3. □ A caterpillar is a tree T with the property that the removal of the end-vertices of T results in a path. This path is referred to as the spine of the caterpillar. If the spine is trivial, the caterpillar is a star; if the spine is K 2 , then the caterpillar is a double star. The next result includes caterpillars whose spine is both trivial and K 2 . Realize that the way of defining the spine provided in the proof is intended to make the vertex labeling easy to describe. Theorem 4. For every caterpillar T of order p, str (T ) = p + 1. Proof. In light of Theorems 2 and 3, it suffices to prove the theorem for caterpillars whose spine contains at least 3 vertices and both end-vertices of the spine have end-vertices attached. The lower bound for str (T ) is obtained immediately from Lemma 2. Please cite this article as: R. Ichishima, F.A. Muntaner-Batle and A. Oshima, On the strength of some trees, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.06.002.
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To establish the upper bound for str (T ), let T be the caterpillar with (m ) } ⋃ { j⏐ V (T ) = {u i |i ∈ [1, m] } ∪ vi ⏐ j ∈ [1, n i ] i=1
and (m ) } ⋃{ j ⏐⏐ E (T ) = {u i u i+1 |i ∈ [1, m − 1] } ∪ u i vi j ∈ [1, n i ] . i=1
Then T has the spine S with V (S) = {u i |i ∈ [1, m] } and E (S) = {u i u i+1 |i ∈ [1, m − 1] } . Note that if we define a permutation π of V (S) by π (u i ) = x j (i ∈ [1, m] and j ∈ [1, m]) , where n ′j = degT u i − deg S u i = n i with n ′1 ≥ n ′2 ≥ · · · ≥ n ′m , then π is an involutive automorphism of S, and we have V (S) = {xi |i ∈ [1, m] }. Hence, every edge of S joins vertices xs and xt with j = i + 1 and i ∈ [1, m − 1] whenever π −1 (xs ) = u j and π −1 (xt ) = u i or π −1 (xs ) = u i and π −1 (xt ) = u j . In the case that n s = n t with 1 ≤ s < t ≤ m, we may assume that π (u s ) is located to the left of π (u t ) in the drawing of T in the plane. The preceding construction allows us to define the caterpillar T with (m ) } ⋃ { j⏐ [ ] V (T ) = {xi |i ∈ [1, m] } ∪ yi ⏐ j ∈ 1, n i′ i=1
and (m ) ⋃ { j⏐ ⏐ −1 } [ ]} −1 ′ xi yi ⏐ j ∈ 1, n i . E (T ) = xs xt ⏐π (xs ) π (xt ) ∈ E (S) ∪ {
i=1
[ ] If we let σi = k=1 n ′k , then p = m + σm . Notice that if n i′ = 0 for some i ∈ [1, m], then we treat 1, n i′ as the empty set. Thus, assume that n i′ ̸= 0 for some i ∈ [1, m]; otherwise, S ∼ = Pm and the result follows from Theorem 1. It remains to show the existence of a numbering f of T for which str f (T ) = p + 1. To complete this, there are two cases to proceed according to the possible values for the integer m. ∑i
Case 1: Assume that m ≤ ⌊ p/2⌋, and consider the labeling f : V (T ) → [1, p] such that { i if v = xi and i ∈ [1, m] , [ ] f (v) = j p − σi + j if v = yi , i ∈ [1, m] and j ∈ 1, n i′ . Then f assigns distinct elements of the set [1, p] to the vertices of T . In particular, f assigns distinct elements of the set [1, m] to the vertices of S. This implies that max { f (u) + f (v) |uv ∈ E (S) } ≤ m + (m − 1) = 2m − 1 ≤ 2 ⌊ p/2⌋ − 1 < p. However, max { f (u) + f (v) |uv ∈ E (T ) \E (S) } = max E i j , Please cite this article as: R. Ichishima, F.A. Muntaner-Batle and A. Oshima, On the strength of some trees, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.06.002.
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⏐ ]} [ where E i j = p − σi + i + j ⏐i ∈ [1, m] and j ∈ 1, n i′ . At this point, let i be the integer so that n i′ ̸= 0 with i ∈ [1, m]. Then n ′1 ≥ n ′2 ≥ · · · ≥ n i′ ≥ 1, which implies that σi−1 ≥ i − 1. Thus, {
max { f (u) + f (v) |uv ∈ E (T ) \E (S) } ≤ max E i j ≤ p − σi + i + n i′ = p − σi−1 + i ≤ p − (i − 1) + i = p + 1. Consequently, max { f (u) + f (v) |uv ∈ E (T ) } ≤ p + 1, so f has the property that str f (T ) = p + 1. ⏐{ ⏐ }⏐ Case 2: Assume that m ≥ ⌊ p/2⌋ + 1, and let l = ⏐ n i′ ⏐i ∈ [1, m] and n i′ = 0 ⏐. Then n i′ ≥ 1 for i ∈ [1, m − l] and n i′ = 0 for i ∈ [m − l + 1, m]. With this knowledge in hand, consider the labeling f : V (T ) → [1, p] such that ⎧ if v = xi and i ∈ [1, m − l] , ⎪ ⎪ i ⎨ m + 1 − i if v = xm−l−1+2i and i ∈ [1, ⌈l/2⌉] , f (v) = m −l +i if v = xm−l+2i and i ∈ [1, ⌊l/2⌋] , ⎪ ⎪ [ ] ⎩ j p − σi + j if v = yi , i ∈ [1, m − l] and j ∈ 1, n i′ . Then f assigns distinct elements of the set [1, p] to the vertices of T . Thereby, f assigns distinct elements of the set [1, m] to the vertices of S as follows: { f (xi ) |i ∈ [1, m − l] } = [1, m − l] , { f (xm−l−1+2i ) |i ∈ [1, ⌈l/2⌉] } = [m − ⌈l/2⌉ + 1, m] , { f (xm−l+2i ) |i ∈ [1, ⌊l/2⌋] } = [m − l + 1, m − ⌈l/2⌉] . For the rest of the proof, we will examine the induced edge labels given by f (u) + f (v) for each uv ∈ E (T ). Then, in a similar manner to Case 1, we have max { f (u) + f (v) |uv ∈ E (T ) \E (S) } ≤ max E i j ≤ p − σi + i + n i′ = p − σi−1 + i ≤ p − (i − 1) + i = p + 1, ⏐ [ ]} where E i j = p − σi + i + j ⏐i ∈ [1, m − l] and j ∈ 1, n i′ . However, there are three subcases to pursue according to the possibility for the edge xs xt ∈ E (S). ⏐ { } Subcase 2.1: Let W1 = xs xt ∈ E (S) ⏐n ′ ̸= 0 and n ′ ̸= 0 . Then {
s
t
max { f (u) + f (v) |uv ∈ W1 } = f (xm−l−1 ) + f (xm−l ) = (m − l − 1) + (m − l) = 2m − 2l − 1. However, since T has ( p − m) end-vertices and (m − l) vertices of S have end-vertices, it follows that p−m ≥ m−l, that is, l ≥ 2m − p.
(1)
This implies that 2m − 2l − 1 ≤ 2 p − 2m − 1.
(2)
Since m ≥ ⌊ p/2⌋ + 1, it also follows from (2) that 2 p − 2m − 1 ≤ 2 p − 2 ⌊ p/2⌋ − 3.
(3)
Please cite this article as: R. Ichishima, F.A. Muntaner-Batle and A. Oshima, On the strength of some trees, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.06.002.
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Thus, the fact that 2 ⌊ p/2⌋ ≥ p − 1 together with (3) implies that max { f (u) + f (v) |uv ∈ W1 } ≤ 2 p − 2 ⌊ p/2⌋ − 3 ≤ p − 2. ⏐ { } Subcase 2.2: Let W2 = xs xt ∈ E (S) ⏐n ′s ̸= 0 and n ′t = 0 . Then max { f (u) + f (v) |uv ∈ W2 } = f (vm−l ) + f (vm−l+1 ) = (m − l) + m = 2m − l. Thus, it follows from (1) that max { f (u) + f (v) |uv ∈ W2 } ≤ 2m − l ≤ 2m − (2m − p) = p. ⏐ } { Subcase 2.3: Let W3 = xs xt ∈ E (S) ⏐n ′s = 0 and n ′t = 0 . Then max { f (u) + f (v) |uv ∈ W3 } = f (xm−l−1+2i ) + f (xm−l+2i ) = (m − l + i) + (m + 1 − i) = 2m − l + 1 for each i ∈ [1, ⌊l/2⌋]. Thus, it follows from (1) that max { f (u) + f (v) |uv ∈ W3 } ≤ 2m − l + 1 ≤ 2m − (2m − p) + 1 = p + 1. Consequently, max { f (u) + f (v) |uv ∈ E (T ) } ≤ p + 1, so f has the property that str f (T ) = p + 1.
□
3. The strength of complete n-ary trees The k-level complete n-ary tree Tn,k is a tree in which the ith level consists of n i−1 vertices and each vertex in level i < k has n ‘sons’ at level i + 1. If n = 2, then Tn,k is referred to as a k-level complete binary tree. The strength of T2,k depends on k as the next result indicates. The proof involves the concept of distance in a graph. For a connected graph G and pair u, v ∈ V (G), the distance d (u, v) between u and v is the length of a shortest u − v path in G. Theorem 5. For every integer k ≥ 2, ( ) str T2,k = 2k . ⏐ ( ( ) )⏐ k ⏐ ⏐ Proof. ( ) The lower bound for str T2,k follows immediately from Lemma 2, since V T2,k = 2 − 1 and δ T2,k = 1. ( ) To the upper bound for str T2,k , it suffices to show the existence of a numbering f of T2,k for which ( establish ) str f T2,k = 2k . Let T2,k be given as a rooted tree in the plane. Then T2,k has a unique vertex of degree 2, so chose this distinct vertex to be the root of T2,k and denoted it by r . Of course, if T2,k is drawn in the plane, then by letting { ( ) } Si = v ∈ V T2,k |i = d (r, v) + 1 , the vertices in each level i ≥ 2 can be ordered from left to right using the integers 1, 2, . . . , |Si |. Since the ith level consists of 2i vertices, it follows that |Si | = 2i−1 . Thus, Si can be written as { ⏐ [ ]} j Si = vi ⏐ j ∈ 1, 2i−1 , Please cite this article as: R. Ichishima, F.A. Muntaner-Batle and A. Oshima, On the strength of some trees, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.06.002.
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where i represents the level that the vertex occupies in T2,k and j represents the place that the vertex in level i occupies in the level starting to count from left to right. If we let { ( ) } E i = uv ∈ E T2,k |u ∈ Si and v ∈ Si+1 , then |E i | = 2i and the edges of E i can also be ordered from left to right in ascending order starting with 1 and j ending up with 2i for each set E i . Thus, we will write ei for the edge in the set E i that occupies position j when j j counting the edges of E i from left to right. This implies that if ei , eil ∈ E i with j < l, then ei is located to the left of eil in the drawing of T2,k in the plane. ( ) ( ) 4 )that str T2,3 =] Since T2,2 ∼ = P3 , it follows from Theorem 1 that str T2,2 = 22 . It also follows from Theorem ( [ 23 , since T2,3 is a caterpillar. Let k be an integer with k ≥ 4, and consider the labeling f : V T2,k → 1, 2k − 1 such that ⎧ [ ] k−1 − 2i + j if i ∈ [1, k − 2] and j ∈ 1, 2i−1 , ⎨ 2 ( ) ⎪ [ ] j 2k−2 + 1 − j if i = k − 1 and j ∈ 1, 2k−2 , f vi = [ ] ⎪ ⎩ 2k−1 − 1 + j if i = k and j ∈ 1, 2k−1 . This implies that [ ] { f (v) |v ∈ Sk−1 } = 1, 2k−2 , [ ] { f (v) |v ∈ Si and i ∈ [1, k − 2] } = 2k−2 + 1, 2k−1 − 1 , [ ] { f (v) |v ∈ Sk } = 2k−1 , 2k − 1 . [ k ] ∑k |Si | = 2k − 1, it follows that f assigns distinct elements of the Since i=1 ( set ) 1, 2 − 1 to the vertices of T2,k . ( ) j j For each ei = uv ∈ E T2,k , where u ∈ Si and v ∈ Si+1 , we write f ei for the induced edge label given by ( ) [ ] ( ) j f (u) + f (v). Notice then that for each j, l ∈ 1, 2i−1 with j < l, we have f ei ≤ f eil if i ∈ [1, k − 1] with ( ) ( ) j i ̸= k − 2, whereas we have f ei ≥ f eil if i = k − 2. Thus, f has the property that { ( )⏐ [ ]} j ⏐ max f ei ⏐i ∈ [1, k − 2] and j ∈ 1, 2i−1 { ( ) ( ) ( 1 ) ( 1 )} = max f v11 + f v22 , f vk−2 + f vk−1 ( ) ( ) = f v11 + f v22 ( ) ( ) = 2k−1 − 1 + 2k−1 − 2 = 2k − 3 and { ( )⏐ ( k−1 ) ( k−2 ) [ ]} ⏐ j 2 max f ek−1 ⏐ j ∈ 1, 2i−1 = f vk−1 + f vk2 ( ) = 1 + 2k−1 − 1 + 2k−1 = 2k , implying that ⏐ ( ) { ( )} str f T2,k = max f (u) + f (v) ⏐uv ∈ E T2,k = 2k . ( ) Consequently, str T2,k = 2k . □ The strength of Tn,k is also determined by considering a drawing of Tn,k as a rooted tree in the plane; the proof is quite similar to that of Theorem 5 and will not be included. Theorem 6. For every two integers n ≥ 2 and k ≥ 2, ( ) nk − 1 + 1. str Tn,k = n−1 Please cite this article as: R. Ichishima, F.A. Muntaner-Batle and A. Oshima, On the strength of some trees, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.06.002.
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R. Ichishima, F.A. Muntaner-Batle and A. Oshima / AKCE International Journal of Graphs and Combinatorics xxx (xxxx) xxx
4. The strength of the corona of graphs The corona of two graphs was introduced by Frucht and Harary [13]. The corona G ⊙ H of two graphs G and H is defined as the graph obtained by taking one copy of G (which has order p) and p copies of H , and then joining the ith vertex of G to every vertex in the ith copy of H . It is now possible to present the following result. Theorem 7. Let G be a graph of order p with δ (G) ≥ 1. If str (G) = p + δ (G), then str (G ⊙ n K 1 ) = (n + 1) p + 1 for every positive integer n. Proof. Suppose that G is a graph of order p with δ (G) ≥ 1, and let f be a strength labeling of G with str (G) = p + δ (G). Assume that f has the property that f (vi ) = i for each i ∈ [1, p], where V (G) = {vi |i ∈ [1, p] }. Further, let H ∼ = G ⊙ n K 1 , and define the graph H with } { ⏐ j V (H ) = V (G) ∪ wi ⏐i ∈ [1, p] and j ∈ [1, n] and { } j⏐ E (H ) = E (G) ∪ vi wi ⏐i ∈ [1, p] and j ∈ [1, n] . Then |V (H )| = (n + 1) p and δ (H ) = 1. Thus, the lower bound for str (H ) follows from Lemma 2. To establish the upper bound for str (H ), consider the labeling g : V (H ) → [1, (n + 1) p] such that { i if x = vi and i ∈ [1, p] , g (x) = j (n + 1) p − ni + j if x = wi , i ∈ [1, p] and j ∈ [1, n] . Then {g (vi ) |i ∈ [1, p] } = [1, p] and } { ( )⏐ j ⏐ g wi ⏐i ∈ [1, p] and j ∈ [1, n] = [ p + 1, (n + 1) p] . This implies that g assigns distinct elements of the set [1, (n + 1) p] to the vertices of H . It is now immediate that p + δ (G) ≤ p + ( p − 1) = 2 p − 1 < (n + 1) p + 1 for every positive integer n. Moreover, g has the property that strg (H ) = max {g (u) + g (v) |uv ∈ E (H ) } ( ) = g w1n + g (v1 ) = (n + 1) p + 1. Consequently, str (H ) = (n + 1) p + 1.
□
The preceding result is of particular interest, since there are infinitely many graphs G for which str (G) = |V (G)| + δ (G) (see [8] for a detailed list of graphs). Applying it with such classes of graphs repeatedly, we obtain other classes of graphs H for which str (H ) = |V (H )| + 1 again. 5. Conclusions In this paper, we have shown that every caterpillar and k-level complete n-ary tree attains the bound provided in Lemma 2. We have also established a formula for the strength of the corona of certain graphs and an arbitrary number of isolated vertices (see Theorem 7). A new tree can be created by taking the corona with isolated vertices. Therefore, applying Theorem 7 with the classes of trees studied in this paper, we obtain other classes of trees T for which str (T ) = |V (T )| + 1. These trees include a class of lobsters (a lobster is a tree T with the property that the removal of the end-vertices of T results in a caterpillar). This motivates us to propose the following problem. Problem 1. For every lobster T , determine the exact value of str (T ). Please cite this article as: R. Ichishima, F.A. Muntaner-Batle and A. Oshima, On the strength of some trees, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.06.002.
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Please cite this article as: R. Ichishima, F.A. Muntaner-Batle and A. Oshima, On the strength of some trees, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.06.002.