A new labeling construction from the -product

ANEWLABELINGCONSTRUCTIONFROMTHE⊗h-PRODUCT S. C. L´ OPEZ, F. A. MUNTANER-BATLE, AND M. PRABU Abstract. The ⊗h-product that refers the title was introduced in 2008 as a generalization of the Kronecker product of digraphs. Many relations among labelings have been obtained since then, always using as a second factor a family of super edge-magic graphs with equal order and size. In this paper, we introduce a new labeling construction by changing the role of the factors. Using this new construction the range of applications grows up considerably. In particular, we can increase the information about magic sums of cycles and crowns. Key Words: Edge-magic, super edge-magic, ⊗h-product, magic sum 2010 Mathematics Subject Classification: Primary 05C78, Secondary 05C76 1. Introduction For the graph theory terminology and notation not defined in this paper we refer the reader to either one of the following sources [3, 4, 8, 21]. However, in order to make this paper reasonably self-contained, we mention that by a (p, q)-graph we mean agraphoforderpand size q. For integers m≤n,wedenoteby[m, n]theset {m, m +1,...,n}. In 1970, Kotzig and Rosa [11] introduced the concepts of edgemagic graphs and edge-magic labelings as follows: Let Gbe a (p, q)-graph. Then G is called edge-magic if there is a bijective function f:V(G)∪E(G)→[1,p+q]such that the sum f(x)+f(xy)+f(y)=kfor any xy ∈E(G). Such a function is called an edge-magic labeling of Gand kis called the valence [11] or the magic sum [21] of the labeling f. We write val(f) to denote the valence of f. Motivated by the concept of edge-magic labelings, Enomoto et al. [5] introduced in 1998 the concepts of super edge-magic graphs and labelings as follows: Let f: V(G)∪E(G)→[1,p+q] be an edge-magic labeling of a (p, q)-graph G with the extra property that f(v) = [1,p].Then G is called super edge-magic and fis a super edge-magic labeling of G. It is worthwhile mentioning that Acharya and Hegde had already defined in [1] the concept of strongly indexable graph that turns out to be equivalent to the concept of super edge-magic graph. Although the original definitions of (super) edge-magic graphs and labelings were originally provided for simple graphs, in this paper, we understand these definitions for any graph without multiple edges. Therefore, unless otherwise specified, the graphs considered in this paper are not necessarily simple. In [6], Figueroa-Centeno et al. provided the following useful characterization of super edge-magic simple graphs, that works in exactly the same way for graphs in general. 1 2S.C.L ´ OPEZ, F. A. MUNTANER-BATLE, AND M. PRABU Lemma 1.1. [6] Let Gbe a (p, q)-graph. Then Gis super edge-magic if and only if there is a bijective function g:V(G)−→ [1,p]such that the set S={g(u)+g(v): uv ∈E(G)}is a set of qconsecutive integers. In this case, gcan be extended to a super edge-magic labeling fwith valence p+q+ min S. Let fbe an edge-magic labeling of a (p, q)-graph G.Thecomplementary labeling of f,denotedbyf, is the labeling defined by the rule: f(x)=p+q+1−f(x), for all x∈V(G)∪E(G). Notice that, if fis an edge-magic labeling of G,wehavethat fis also an edge-magic labeling of Gwith valence val(f)=3(p+q+1)−val(f). In the case of a super edge-magic labeling fof a graph G, there is also the corresponding super edge-magic complementary labeling,fc, which is also super edge-magic. In this case fcis defined by the rule fc(x)=p+1−f(x),∀x∈V(G),(1) fc(ab) is obtained as described in Lemma 1.1, for all ab ∈E(G). Then, the valence of fccan be expressed in terms of the valence of fas follows: val(fc)=4p+q+3−val(f). Also, in the case that Gis a graph of equal order and size, new edge-magic labelings can be obtained from known super edge-magic labelings of G. Such labelings are known as the even and odd edge-magic labelings of G.Theodd labeling and the even labeling [17] obtained from f, denoted respectively by o(f)ande(f), are the labelings o(f),e(f):V(G)∪E(G)→[1,p+q] defined as follows: (i) on the vertices: o(f)(x)=2f(x)−1ande(f)(x)=2f(x), for all x∈V(G), (ii) on the edges: o(f)(xy)= 2val(f)−2p−2−o(f)(x)−o(f)(y)ande(f)(xy)=2val(f)−2p−1−e(f)(x)−e(f)(y), for all xy ∈E(G). The next lemma is an easy exercice. Lemma 1.2. Let fbe a super edge-magic labeling of a (p, q)-graph G.Then, e(f)≃o(fc)and o(f)≃e(fc). In [7], Figueroa et al. defined the following product: Let Dbe a digraph and let Γbe a family of digraphs with the same set Vof vertices. Assume that h:E(D)→Γis any function that assigns elements of Γto the arcs of D. Then the digraph D⊗hΓis defined by (i) V(D⊗hΓ) = V(D)×Vand (ii) ((a, i),(b, j)) ∈E(D⊗hΓ)⇔(a, b)∈E(D)and (i, j)∈E(h(a, b)). Note that when his constant, D⊗hΓis the Kronecker product. Many relations among labelings have been established using the ⊗h-product and some particular families of graphs, namely Spand Sk p(see for instance, [10, 15, 18, 19]). The family Spcontains all super edge-magic 1-regular labeled digraphs of order pwhere each vertex takes the name of the label that has been assigned to it. A super edgemagic digraph Fis in Sk pif |V(F)|=|E(F)|=pand the minimum sum of the labels of the adjacent vertices is equal to k(see Lemma 1.1). Notice that, since each 1-regular digraph has minimum induced sum equal to (p+3)/2, Sp⊂S (p+3)/2 p. The following result was introduced in [18], generalizing a previous result found in [7] : Theorem 1.1. [18] Let Dbe a (super) edge-magic digraph and let h:E(D)→S k pbe any function. Then D⊗hSk pis (super) edge-magic. In this paper, we characterize some relations among the induced labelings obtained from the ⊗h-product, when we combine the odd and the even labelings of a particular ANEWLABELINGCONSTRUCTIONFROMTHE⊗h-PRODUCT 3 super edge-magic labeling f, together with the complementary and the super edgemagic complementary constructions of the labelings involved. This is the content of Section 2. The main result of the paper is Theorem 3.1, where, in some sense, we exchange the role of the factors established in Theorem 1.1. Thus, we can enlarge the family of labeled graphs that we can obtain from the product. We conclude the paper with an application of this fact in Section 4. 2. Some labeling properties obtained from the ⊗h-product One of the research lines when we deal with edge-magic labelings of a particular graph Gis the study of the theoretical valences that are realizable. This problem has been completely solved for crowns of the form Cm⊙Kn,wherem=pkand m=pq, and pand qare primes (see [16, 17] and [14], respectively). In both cases, the proof is based in the construction of all theoretical super edge-magic valences and then, with the help of the odd and the even labelings, to complete the remaining valences. In this section we show some labeling properties in which we combine these labelings together with other labeling constructions. The key point in the proof of Theorem 1.1 is to rename the vertices of Dand each element of Sk pafter the labels of their corresponding (super) edge-magic labeling fand their super edge-magic labelings respectively and to define the labels of the product as follows: (i) the vertex (a, i)∈V(D⊗hSk p) receives the label: p(a−1) + iand (ii) the arc ((a, i),(b, j)) ∈E(D⊗hSk p) receives the label: p(e−1) + (k+p)−(i+j), where e is the label of (a, b) in D. Thus, for each arc ((a, i),(b, j)) ∈E(D⊗hSk p), coming from an arc e=(a, b)∈E(D)andanarc(i, j)∈E(h(a, b)), the sum of labels is constant and equal to p(a+b+e−3) + (k+p). That is, p(val(f)−3) + k+p.Thus,weget the next result. Lemma 2.1. [18] Let ˆ fbe the (super) edge-magic labeling of the graph D⊗hSk pinduced by a (super) edge-magic labeling fof D.Thenthevalenceofˆ fis given by the formula val(ˆ f)=p(val(f)−3) + k+p. (2) The following proposition shows a relation among complementary labelings and induced labelings obtained from the ⊗h-product. Proposition 1. Let fbe an edge-magic labeling of a digraph D.Consideranyfunction h:E(D)→S k p.Then,thereexists¯ h:E(D)→S p+3−k psuch that D⊗hSk p∼ =D⊗¯ hSp+3−k pand ˆ f≃ˆ f, where ˆ fis the complementary labeling of the induced labeling of fof D⊗hSk pand ˆ fis the labeling of D⊗¯ hSp+3−k pinduced by the complementary labeling of f. Proof. We will prove that the induced edge-magic labeled digraphs are isomorphic. Let φ:Sk p→S p+3−k pbe the function defined by φ(F)=Fc,where( ¯ i, ¯ j)∈Fcif and only if (p+1−¯ i, p +1−¯ j)∈F. Notice that the minimum induced edge sum of Fcis 2p+2−(i+j), where i+jis the maximum induced edge sum of F, that is, i+j=k+(p−1). Thus, the minimum induced edge sum of Fcis (p+3)−k. 4S.C.L ´ OPEZ, F. A. MUNTANER-BATLE, AND M. PRABU Assume that Dis a (n, m)-digraph in which each vertex is identified with the label assigned to it by f. Then, the induced labeling ˆ fof the product D⊗hSk pis defined by ˆ f(a, i)=p(a−1) + i,foranyvertex(a, i)∈V(D⊗hSk p)and ˆ f((a, i),(b, j)) = p(e−1) + k+p−(i+j), where eis the label of (a, b) assigned by f. Then, since |V(D⊗hSk p)|=pn and |E(D⊗hSk p)|=pm, the complementary labeling of ˆ fis defined by -ˆ f(a, i)=p(m+n)+1−p(a−1) −i,foranyvertex(a, i)∈V(D⊗hSk p)and -ˆ f((a, i),(b, j)) = p(m+n)+1−p(e−1) −k−p+(i+j), where eis the label of (a, b) assigned by f. Let ¯ h=φ◦h:E(D)→S p+3−k pand consider the labeling ¯ fof D. Then the induced labeling ˆ fof the product D⊗¯ hSp+3−k pis defined by ˆ f(¯a,¯ i)=p(m+n+1−a−1)+p+1−i, that is, -ˆ f(¯a,¯ i)=p(m+n)+1−p(a−1) −i,foranyvertex(a, i)∈V(D⊗hSk p)and -ˆ f((¯a,¯ i),(¯ b, ¯ j)) = p(m+n+1−e−1) + p+3−k+p−(¯ i+¯ j), where eis the label of (a, b) assigned by f. That is, ˆ f((¯a,¯ i),(¯ b, ¯ j)) = p(m+n)+1−p(e− 1) −k−p+(i+j). This proves the result. ! Corollary 2.1. Let Dbe a (super) edge-magic digraph. Let fand fbe a (super) edge-magic labeling and its complement of Drespectively. Assume that k=(p+3)/2 and let ˆ fand ˆ fbe the edge-magic labeling and its complementary labeling ofthegraph und(D⊗hSp+3 p)obtained from the labeling fof D. Then, val(ˆ f)=val(ˆ f). Proof. It sufficies to observe that if k=(p+3)/2thenp+3−k=(p+3)/2. ! For digraphs Dwith the same order and size, we obtain the next two results. Corollary 2.2. Let fbe a super edge-magic labeling of a (n, m)-digraph Dwith m=n. Consider any function h:E(D)→S k p.Then,thereexists ¯ h:E(D)→S p+3−k psuch that ! o(f)≃! e(fc), where ! o(f)is the complementary labeling of the induced labeling of o(f)of D⊗hSk p and ! e(fc)is labeling of D⊗¯ hSp+3−k pinduced by the even labeling of fc. Proof. Let fbe a super edge-magic labeling of D, by Proposition 1, there exists ¯ h: E(D)→S p+3−k psuch that ! o(f)≃! o(f), where ! o(f) is the complementary labeling of the induced labeling of o(f)ofD⊗hSk p and ! o(f) is labeling of D⊗¯ hSp+3−k pinduced by the complementary of the odd labeling of f. By Lemma 1.2, o(f)≃e(fc). Thus, we obtain the result. ANEWLABELINGCONSTRUCTIONFROMTHE⊗h-PRODUCT 5 ! With a similar proof, we obtain the next corollary. Corollary 2.3. Let fbe a super edge-magic labeling of (n, m)-digraph Dwith m=n. Consider any function h:E(D)→S k p.Then,thereexists ¯ h:E(D)→S p+3−k psuch that ! e(f)≃! o(fc), where ! e(f)is the complementary labeling of the induced labeling of e(f)of D⊗hSk p and ! o(fc)is labeling of D⊗¯ hSp+3−k pinduced by the odd labeling of fc. The next result is similar to Proposition 1. Proposition 2. Let fbe a super edge-magic labeling of digraph D. Consider any function h:E(D)→S k p.Then,thereexists¯ h:E(D)→S p+3−k psuch that (ˆ f)c≃" fc, where (ˆ f)cis the super edge-magic complementary labeling of the induced labeling of fof D⊗hSk pand " fcis the labeling of D⊗¯ hSp+3−k pinduced by the super edge-magic complementary labeling of f.Moreover,val(( ˆ f)c)=val(" fc). Proof. Let φ:Sk p→S p+3−k pbe the function defined by φ(F)=Fc,where( ¯ i,¯ j)∈Fc if and only if (p+1−¯ i, p +1−¯ j)∈F. Notice that the minimum induced edge sum of Fcis 2p+2−(i+j), where i+jis the maximum induced edge sum of F, that is, i+j=k+(p−1). Thus, the minimum induced edge sum of Fcis (p+3)−k. Assume that Dis a (n, m)-digraph in which each vertex is identified with the label assigned to it by f. Then, the induced (super edge-magic) labeling ˆ fof the product D⊗hSk pis defined by ˆ f(a, i)=p(a−1) + i,foranyvertex(a, i)∈V(D⊗hSk p). Since |V(D⊗hSk p)|=pn, the super edge-magic complementary labeling of ˆ fis defined by -( ˆ f)c(a, i)=pn +1−(p(a−1) + i), for any vertex (a, i)∈V(D⊗hSk p). Let ¯ h=φ◦h:E(D)→S p+3−k pand consider the labeling fcof D. Then the induced labeling " fcof the product D⊗¯ hSp+3−k pis defined by " fc(¯a,¯ i)=p(¯a−1) +¯ i, that is, -" fc(¯a,¯ i)=p(n−a)+p+1−i,foranyvertex(a, i)∈V(D⊗hSk p) This proves the result. ! 3. The main result Since the ⊗h-product was first introduced in 2008 [7], it has been proven to be an excellent technique to better understand many different types of labelings, as for instance (super) edge-magic labelings and harmonious labelings. The lack of enumerative results involving graph labelings constitutes a big gap in the literature of graph labelings that this product has helped to fill enormously. Also further applications outside the world of graph labeling have been found for the ⊗h-product, as for instance it introduces new ways to construct Skolem and Langford type sequences [13]. In summary, 6S.C.L ´ OPEZ, F. A. MUNTANER-BATLE, AND M. PRABU the ⊗-product constitutes a big breakthru into the world of graph labeling that allows to have a better and deeper understanding of the subject. In all the results involving the ⊗h-product, since the very beginning, it seems to be a constant to use super edge-magic labeled graphs as the second factor of the product, or at least graphs that in a way or another come from super edge-magic graphs [10, 15, 18]. The power of this section lies in the fact that it allows us to use other types of labeled graphs as a second factor of the product and this allows to refresh the ways of attacking old famous problems in the subject of graph labelings as we will in the next lines. We now introduce a new family Tq σof edge-magic labeled graphs. An edge-magic labeled digraph Fis in Tq σif V(F)=V,|E(F)|=qand the magic sum of the edgemagic labeling is equal to σ. Theorem 3.1. Let D∈S k nand let hbe any function h:E(D)→Tq σ.ThenD⊗hTq σ is edge-magic. Proof. Let p=|V|. We identify the vertices of Dand each element of Tq σafter the labels of their corresponding super edge-magic labeling and edge-magic labeling, respectively. Consider the following labeling of D⊗hTq σ: (1) If (i, a)∈V(D⊗hTq σ) we assign to the vertex the label: (p+q)(i−1) + a. (2) If ((i, a),(j, b)) ∈E(D⊗hTq σ) we assign to the arc the label: (p+q)(k+n−(i+j)−1) + (σ−(a+b)). Notice that, since D∈S k nis labeled with a super edge-magic labeling with minimum sum of the adjacent vertices equal to k,wehave {(k+n)−(i+j): (i, j)∈E(D)}= [1,n]. Moreover, since each element F∈Tq σ, it follows that {(σ−(a+b): (a, b)∈E(F)}= [1,p+q]\V. Thus, the set of labels in D⊗hTq σcovers all elements in [1,n(p+q)]. Moreover, for each arc ((i, a)(j, b)) ∈E(D⊗hTq σ) the sum of the labels is constant and is equal to: (p+q)(k+n−3) + σ.! From the previous proof, we also conclude the next result. Lemma 3.1. Let D∈S k nand # hbe the edge-magic labeling of the graph D⊗hTq σ, induced by the super edge-magic labeling of Dand the function h:E(D)→Tq σ.Then the valence of # his given by the formula val(# h)=(p+q)(k+n−3) + σ,(3) where p=|V(F)|, for every F∈Tq σ. ANEWLABELINGCONSTRUCTIONFROMTHE⊗h-PRODUCT 7 3.1. More labeling properties obtained from the ⊗h-product. Recall that, for every labeled digraph D∈S k nwe can consider Dc∈S n+3−k n,suchthatD∼ =Dc,just by taking the super edge-magic complementary labeling that defines D. Proposition 3. Let D∈S k nand let h:E(D)→T q σbe any function. Then, there exists hc:E(Dc)→Tq 3(p+q+1)−σsuch that D⊗hTq σ≃Dc⊗hcTq 3(p+q+1)−σ,and # hc≃¯ # h, where ¯ # his the edge-magic complementary labeling of the induced labeling of D⊗hTq σ and # hcis the induced labeling of Dc⊗hcTq 3(p+q+1)−σ. Proof. Let φ:Sk n→S n+3−k nbe the function defined by φ(D)=Dc,where( ¯ i, ¯ j)∈Dcif and only if (p+1−¯ i, p +1−¯ j)∈Dand ψ:Tq σ→Tq 3(p+q+1)−σbe the function defined by ψ(F)= ¯ F,where( ¯ i, ¯ j)∈¯ Fif and only if (p+q+1−¯ i, p +q+1−¯ j)∈F. Let h:E(D)→T q σbe any function. Then, the induced edge-magic labeling # h of the product D⊗hTq σis defined by # h(i, a)=(p+q)(i−1) + a,foranyvertex (i, a)∈V(D⊗hTq σ)andby# h((i, a),(j, b)) = (p+q)(k+n−(i+j)−1)+(σ−a−b), for any arc ((i, a),(j, b)) ∈E(D⊗hTq σ). Then, since |V(D⊗hTq σ)|=pn and |E(D⊗hTq σ)|=qn, the complementary labeling ¯ # hof D⊗hTq σis defined by -¯ # h(i, a)=(p+q)n+1−(p+q)(i−1) −a, that is, ¯ # h(i, a)=(p+q)(n+1−i−1) + (p+q+1−a), for any vertex (i, a)∈V(D⊗hTq σ)and -¯ # h((i, a),(j, b)) = (p+q)n+1−(p+q)(k+n−(i+j)−1) −(σ−a−b), that is, ¯ # h((i, a),(j, b)) = (p+q)(n+3−k+n−(n+1−i)−(n+1−j)−1) +3(p+q+1)−σ−(p+q+1−a)−(p+q+1−b). Thus, the function hc:E(Dc)→Tq 3(p+q+1)−σdefined by hc(i, j)=ψ(h(n+1−i, n +1−j)), induces a labeling # hcof Dc⊗hcTq 3(p+q+1)−σ, which is isomorphic to the labeling ¯ # hof D⊗hTq σ. Therefore, the result follows. ! 4. Magic sums of cycles A famous conjecture of Godbold and Slater [9] states that, for n=2t+1≥7and 5t+4≤j≤7t+5andforn=2t≥4and5t+2≤j≤7t+ 1 there is an edge-magic labeling of Cn, with valence k=j. Let Gbe a (p, q)-graph and f:V(G)∪E(G)→[1,p+q] be a bijective function. The f-weight of a vertex v∈V(G), wf(v), is defined to be wf(v)=f(v)+$f(e), where the sum is taken over all edges eincident to v. The function fis said to be a vertex-magic total labeling [20], if the vertex weight wtf(v)doesnotdependonv.Itturnsout,that for 2-regular graphs the notions of edge-magic labeling and vertex-magic total labeling coincide, since we can easily obtain a vertex-magic total labeling from an edge-magic 8S.C.L ´ OPEZ, F. A. MUNTANER-BATLE, AND M. PRABU labeling and viceversa, just by translating one unit clockwise the labels: the label of each edge is assigned to one of its adjacent vertices, and the label of the other one is assigned to the edge. Dan McQuillan proved in [22] the next result that was originally stated in terms of vertex-magic total labelings. Proposition 4. [22] Let pbe odd. Assume that Cmhas an edge-magic labeling f. Then, (i) Cpm has an edge-magic labeling with valence p(val(f)) −3(p−1)/2,and (ii) Cpm has an edge-magic labeling with valence 3(p−1)m+val(f). The following structural results will be useful to prove that Proposition 4 can also be obtained by means of the ⊗h-product. We denote by −→ Cnand by ←− Cnthe two possible strong orientations of the cycle Cn, where the vertices of Cnare the elements of the set {i}n i=1. It is well known that −→ Cm⊗h{−→ Cn,←− Cn}=gcd(m,n)−→Clcm[m,n]. Theorem 4.1. [2] Let m, n ∈Nand consider the product −→Cm⊗h{−→Cn,←−Cn}where h:E(−→Cm)−→ {−→Cn,←−Cn}.Letgbe a generator of a cyclic subgroup of Zn,namely ⟨g⟩,suchthat|⟨g⟩|=k.AlsoletNg(h−)<mbe a natural number that satisfies the congruence relation m−2Ng(h−)≡g(mod n). If the function hassigns ←−Cnto exactly Ng(h−)arcs of −→Cmthen the product −→Cm⊗h{−→Cn,←−Cn} consists of exactly n/k disjoint copies of a strongly oriented cycle −→Cmk.Inparticular if gcd(g,n)=1,then⟨g⟩=Znand if the function hassigns ←−Cnto exactly Ng(h−)arcs of −→Cmthen −→Cm⊗h{−→Cn,←−Cn}∼ =−→Cmn. Corollary 4.1. [17] Let n≥3be an odd integer and suppose that m≥3is an integer such that either mis odd or m≥n.Thenthereexistsafunctionh:E(−→ Cm)→{ −→ Cn,←− Cn} such that −→ Cm⊗h{−→ Cn,←− Cn}∼ =−−→ Cmn. Now, by combining the previous two results and Lemmas 2.1 and 3.1, we obtain the next result, which, except for the technical condition in (i), it is the same result that McQuilian obtained in [22] (see Proposition 4). Proposition 5. Let pbe odd. Assume that Cmhas an edge-magic labeling f.Then, (i) Cpm has an edge-magic labeling with valence p(val(f)) −3(p−1)/2,whenmis odd or m≥p. (ii) Cpm has an edge-magic labeling with valence 3(p−1)m+val(f). Proof. (i) By Corollary 4.1, there exists a function h:E(−→ Cm)→{ −→ Cp,←− Cp}such that −→ Cm⊗h{−→ Cp,←− Cp}∼ =−−→ Cpm.Assume that each vertex of Cpis identified by the label assigned ANEWLABELINGCONSTRUCTIONFROMTHE⊗h-PRODUCT 9 to it by a super edge-magic labeling. Then, by Lemma 2.1, the induced labeling of the product −−→ Cpm has valence: val( ˆ f)=p(val(f)−3) + (p+3)/2+p, that is, p(val(f)) −3(p−1)/2. (ii) Similarly, By Theorem 4.1, there exists a function h:E(−→ Cp)→{ −→ Cm,←− Cm}such that −→ Cp⊗h{−→ Cp,←− Cp}∼ =−−→ Cpm. Assume that each vertex of Cpis identified by the label assigned to it by a super edge-magic labeling and each vertex of Cmis identified by the label assigned to it by f. Then, by Lemma 3.1, the induced labeling of the product −−→ Cpm has valence: val( ˜ f)=2m((p+3)/2+p−3) + val(f), that is, 3(p−1)m+ val(f). Thus, the result holds. ! Acknowledgements The research conducted in this document by the first author has been supported by the Spanish Research Council under project MTM2014-60127-P and symbolically by the Catalan Research Council under grant 2014SGR1147. References [1] B. D. Acharya and S. M. Hegde, Strongly indexable graphs, Discrete Math. 93 (1991), 123– 129. [2] A. Ahmad, F. A. Muntaner-Batle, M. 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