A Multigraph Characterization of Permutiple Strings

#A111 INTEGERS 25 (2025) A MULTIGRAPH CHARACTERIZATION OF PERMUTIPLE STRINGS Benjamin V. Holt Department of Mathematics, Southwestern Oregon Community College, Oregon [email protected] Received: 5/11/25, Accepted: 11/14/25, Published: 11/25/25 Abstract A permutiple is a natural number whose representation in some base is an integer multiple of a number whose representation has the same collection of digits. A previous paper utilizes a finite-state-machine construction and its state graph to recognize permutiples and to generate new examples. Permutiples are associated with walks on the state graph which necessarily satisfy certain conditions. However, the above effort does not provide sufficient conditions for the existence of permutiples. In this paper, we provide such a condition which we will state using the language of multigraphs. 1. Introduction Let b > 1 be an integer. A permutiple is a natural number which is an integer multiple of some permutation of its digits in base b[8]. Specific cases of digitpermutation problems include cyclic permutations of digits [4, 12], for example, 714285 = 5 ·142857,as well as digit reversals [5, 6, 7, 13, 16, 18, 19, 20], which include 87912 = 4 ·21978 and 98901 = 9 ·10989.Numbers which are multiples of cyclic permutations of their digits are known as cyclic numbers [4]. Numbers which are multiples of their reversals are known by several names, including palintiples [5, 6, 7], reverse multiples [13, 16, 19, 20], and reverse divisors [18]. Another specific case of permutiples worth mentioning is a paper by Qu and Curran [14] which considers base-bnumbers which are multiples of (bb−1−1)/(b−1)2, whose representation is all consecutive base-bdigits 1 through b−1.Two base-10 examples include 987654312 = 8 ·123456789 and 493827156 = 4 ·123456789. Some works place no restrictions on the type of permutation which may arise in an arbitrary base and multiplier [8, 9, 10, 11]. In [8, 9], the author describes methods for finding new examples of permutiples from the digits of known examples. For instance, using these methods, we are able to find new examples 79128 = 4 ·19782 DOI: 10.5281/zenodo.17711714 INTEGERS: 25 (2025) 2 and 78912 = 4 ·19728 from the known example mentioned above, 87912 = 4 · 21978. Other works by the author [10, 11] utilize graph-theoretical and finitestate-machine methods to produce examples with any suitable number of digits. These methods are modifications of the work of Hoey [5] and Sloane [16], and both apply what are ultimately similar techniques to the digit-reversal problem. The former uses finite-state-machine methods, while the latter uses a graph-theoretical approach. Another example of applying finite-state-machine techniques to basedependent integer sequences is a paper by Faber and Grantham [2] which finds pairs of integers whose sum is the reverse of their product. For example, 3 and 24 have this property since 3 + 24 = 27 and 3 ·24 = 72. The methods developed in [10] use a finite-state-machine construction, known as the Hoey-Sloane machine, and its state graph, known as the Hoey-Sloane graph, which describes a collection of possible base-bmultiplications by a single-digit multiplier n. The states of the Hoey-Sloane machine are the possible carries which may occur when performing digit-preserving multiplication, and the input alphabet consists of ordered pairs representing directed edges from the mother graph, which catalogs how digits may be permuted in a single-digit multiplication. The initial state of the machine must be zero since the carry corresponding to the first digit in any single-digit multiplication is defined to be zero [8]. Similarly, the terminal digit of a multiplication requires the final (accepting) state to be zero, otherwise there would be digits beyond the terminal digit. It is shown in [10] that input strings which represent permutiples, known as permutiple strings, consist of ordered pairs which make up cycles on the mother graph. In this context, permutiples correspond to a sequence of state transitions (walks on the Hoey-Sloane graph) beginning and ending with the zero state, where each transition is induced by a collection of edges which is a multiset union of cycles of the mother graph. In this paper, we continue the work described above to establish sufficient conditions for the existence of permutiple strings, yielding a multigraph characterization of permutiple strings. We also apply the results to several examples, which describe how to create novel permutiple examples of any suitable length. 2. Summary of Previous Work What follows is a summary of several results and defnitions from previous works [8, 10] that we will use in this article. 2.1. Basic Definitions and Results The notation (dk, dk−1, . . . , d0)bis used to represent Pk j=0 djbj,where 0 ≤dj< b for all 0 ≤j≤k. We may now define what it means to be a permutiple number. INTEGERS: 25 (2025) 3 Definition 1 ([8]).Let 1 < n < b be a natural number, and let σbe a permutation on {0,1,2,...,k}. We say that (dk, dk−1, . . . , d0)bis an (n, b, σ)-permutiple provided (dk, dk−1, . . . , d1, d0)b=n(dσ(k), dσ(k−1), . . . , dσ(1), dσ(0))b. When σitself is not important to the discussion, we will refer to (dk, dk−1, . . . , d0)b as simply an (n, b)-permutiple. The collection of all base-bpermutiples having multiplier nwill be referred to as (n, b)-permutiples. The next result relates the digits and carries of a permutiple. Theorem 1 ([8]).Let (dk, dk−1, . . . , d0)bbe an (n, b, σ)-permutiple, and let cjbe the jthcarry. Then, bcj+1 −cj=ndσ(j)−dj for all 0≤j≤k. The carries in any permutiple multiplication are always less than the multiplier. Theorem 2 ([8]).Let (dk, dk−1, . . . , d0)bbe an (n, b, σ)-permutiple, and let cjbe the jth carry. Then, cj≤n−1for all 0≤j≤k. 2.2. Permutiple Graphs and the Mother Graph For any permutiple, we may define a directed graph which describes the most essential properties of the digit permutation. Definition 2 ([10]).Let p= (dk, dk−1, . . . , d0)bbe an (n, b, σ)-permutiple. We define a directed graph, called the graph of p, denoted by Gp,to consist of the collection of base-bdigits as vertices, and the collection of directed edges Ep= dj, dσ(j)|0≤j≤k.A graph Gfor which there is a permutiple psuch that G=Gpis called a permutiple graph. For the remainder of this paper, we may drop the “directed” terminology with the understanding that all graphs and multigraphs considered here will have directed edges. Table 1 gives a collection of permutiples with the same graph. Their common graph is shown in Figure 1. (4,10, σ)-Permutiple σ (8,7,9,1,2)10 = 4 ·(2,1,9,7,8)10 ρ (8,7,1,9,2)10 = 4 ·(2,1,7,9,8)10 (1,2)ρ(1,2) (7,9,1,2,8)10 = 4 ·(1,9,7,8,2)10 ψ−4ρψ4 (7,1,9,2,8)10 = 4 ·(1,7,9,8,2)10 ψ−4(1,2)ρ(1,2)ψ4 Table 1: Permutiples with the same graph as (8,7,9,1,2)10 = 4 ·(2,1,9,7,8)10, where ψ= (0,1,2,3,4) and ρis the reversal permutation. INTEGERS: 25 (2025) 4 0 1 23 4 5 6 7 8 9 Figure 1: The directed graph which results from taking the collection of ordered pairs dj, dσ(j)|0≤j≤4from any example in Table 1 as edges. Permutiple graphs provide a framework for classifying permutiples. Definition 3 ([10]).Let pbe an (n, b)-permutiple with graph Gp.We define the class of pto be the collection Cof all (n, b)-permutiples qsuch that Gqis a subgraph of Gp.We also define the graph of the class to be Gp,which we will denote as GC and will call the graph of C. For an (n, b)-permutiple (dk, . . . , d0)b=n(dσ(k), .. . , dσ(0))bit is shown in [10] that λ(dj+ (b−n)dσ(j))≤n−1 for all 0 ≤j≤k, where λgives the least nonnegative residue modulo b. This condition puts a restriction on the possible edges of a permutiple graph, and we state it as a theorem. Theorem 3 ([10]).Let p= (dk, dk−1, . . . , d0)bbe an (n, b, σ)-permutiple with graph Gp.Then, for any edge (dj, dσ(j))of Gp,it must be that λdj+ (b−n)dσ(j)≤n−1 for all 0≤j≤k, where λgives the least non-negative residue modulo b. Theorem 3 enables us to gather all possible edges of a permutiple graph into a single graph to obtain the mother graph. Definition 4 ([10]).The (n, b)-mother graph, denoted M, is the graph having all base-bdigits as its vertices and the collection of edges (d1, d2) satisfying the inequality λ(d1+ (b−n)d2)≤n−1. The next result underpins the methods presented in [10]. Theorem 4 ([10]).Let Cbe any (n, b)-permutiple class. Then, GCis a union of cycles of M. We now provide the reader with an example which brings together the concepts we have covered so far. INTEGERS: 25 (2025) 5 Example 1. The (4,10)-mother graph is displayed in Figure 2. Letting p= (8,7,9,1,2)10 = 4 ·(2,1,9,7,8)10 from Table 1, and Cbe the (4,10)-permutiple class with graph GC=Gp,Figure 2 features the graph of Cin bold red. 0 1 23 4 5 6 7 8 9 Figure 2: The (4,10)-mother graph with the graph of Cfeatured in bold red. 2.3. Finite-State-Machine Description of the Permutiple Problem Taking the carries of a permutiple as a collection of states, the above concepts can be placed within a finite-state-machine framework. Taking non-negative integers less than nas the collection of states, and the edges of Mas the input alphabet, the equation c2= [nd2−d1+c1]÷b(1) defines a state-transition function from state c1to state c2with (d1, d2) serving as the input which induces the transition. This transition corresponds to a labeled edge on the state diagram as seen in Figure 3. c1c2 (d1, d2) Figure 3: An edge on the state diagram. The first carry c0of any single-digit multiplication is zero by definition [8]. This is to say that the initial state must be zero. Also, for an ℓ-digit (n, b)-permutiple, cℓ must also be zero, otherwise, the result would be an (ℓ+ 1)-digit number. Thus, INTEGERS: 25 (2025) 6 the zero state is the only possible accepting state. The above is called the (n, b)- Hoey-Sloane machine, and its state diagram is called the (n, b)-Hoey-Sloane graph, which we denote as Γ. Digit pairs (d1, d2) which solve Equation (1) for particular values of c1and c2 are not unique. That is, there are generally multiple inputs that the machine will recognize for a transition to occur. Thus, a collection of inputs is assigned to each edge on Γ by the mapping (c1, c2)7→ {(d1, d2)∈EM|c2= [nd2−d1+c1]÷b}, where EMis the collection of edges of M. In the next section, we will define a labeled multigraph representation of Γ,where a unique multi-edge is assigned to each input. The language of input strings accepted by the (n, b)-Hoey-Sloane machine is denoted as L. Thus, Lmay be described as finite sequences of edge-label inputs which define walks on Γ whose initial and final states are zero. Such walks are called L-walks. Members of Lwhich produce permutiple numbers are called (n, b)- permutiple strings. We may interpret Theorem 4 anew in this setting. Corollary 1 ([10]).Let s= (d0,ˆ d0)(d1,ˆ d1)···(dk,ˆ dk)be a member of L. If sis a permutiple string, then the collection of ordered-pair inputs of sis a union of cycles of M. Corollary 1 tells us that any permutiple string must consist of a collection of mother-graph edge inputs which induce an L-walk on Γ and whose union is a collection of cycles of M. We note, however, that satisfying the above two conditions is not sufficient for a string to be a permutiple string. An example is provided by [10] of a member of Lwhose union is a collection of mother-graph cycles, yet is not a permutiple string. A more precise description of these ideas requires that we restate some definitions from [10]. Definition 5 ([10]).Let C={C0, C1, . . . , Cm}be the cycles of M. For each element Cjof C,define a subgraph Γjof Γ,where each edge of Γjis assigned the edge-label collection by the mapping (c1, c2)7→ {(d1, d2)∈Cj|c2= [nd2−d1+c1]÷b}.Any edge (c1, c2) for which this collection is empty will not be included as an edge on Γj.With the above edges, any state for which both the indegree and outdegree are zero will not be included as a vertex. Each Γjwill be referred to as the image of Cj,or simply as a cycle image. In what follows, we will distinguish the usual set-theoretic union ∪from multiset unions by using the notation ⊎.For example, the multiset union of the multisets {1,2,2,3}and {2,3,4}is denoted as {1,2,2,3} ⊎ {2,3,4}={1,2,2,2,3,3,4}.We suppose that Iis a multiset whose support is a subset Jof {0,1, . . . , m}.Then, if the cycle-image union ΓJ=Sj∈JΓj(edge labels included) is a strongly-connected subgraph of Γ containing the zero state, then ΓJdescribes a machine which recognizes members of Lwhose inputs form the union of cycles Sj∈JCj.If the multiset INTEGERS: 25 (2025) 7 cycle union CI=Uj∈ICjcan be ordered into a string sbelonging to L, then s is a permutiple string. We note that every element of CImust be used, including repeated elements, otherwise, the multisets of left and right components will not be equal, resulting in a multiplication which does not preserve the digits. We now provide the reader with another example by considering the (4,10)- Hoey-Sloane graph. Example 2. The (4,10)-Hoey-Sloane graph is shown in Figure 4. The graph depicting the union of the images of the cycles of GCfrom Example 1 is shown in bold red. The cycles of GCare C0={(9,9)}, C1={(2,8),(8,2)},and C2={(1,7),(7,1)}, and the cycle-image union may be more precisely denoted as Γ0∪Γ1∪Γ2. Using the Hoey-Sloane graph, the multiset union C0⊎C1⊎C1⊎C2⊎C2may be ordered into a member of L, forming a permutiple string. There are multiple ways of accomplishing this, one of which is (8,2)(8,2)(2,8)(9,9)(1,7)(1,7)(7,1)(2,8)(7,1), yielding a new (4,10)-permutiple (7,2,7,1,1,9,2,8,8)10 = 4·(1,8,1,7,7,9,8,2,2)10. Another possible ordering is (8,2)(2,8)(1,7)(7,1)(2,8)(9,9)(1,7)(7,1)(8,2),which gives the new example (8,7,1,9,2,7,1,2,8)10 = 4 ·(2,1,7,9,8,1,7,8,2)10. As noted in [11], not every multiset union can be ordered into a permutiple string. As a trivial example, any of the above cycles individually are not sufficient to form an element of L. A less trivial example is C1⊎C1⊎C2,which is also impossible to order into a member of L. We will address this specific case in a later example. Generally speaking, we see that a strongly-connected union of cycle images containing the zero state is a necessary condition for ordering its corresponding multiset union of mother-graph cycles into permutiple strings [10], but it is not a sufficient one. Finding sufficient conditions is the purpose of this effort. 0 start 1 2 3 (0,0),(4,1),(8,2)(3,3),(7,4) (2,5),(6,6) (1,7),(5,8),(9,9) (2,3),(6,4) (1,0),(5,1),(9,2) (0,2),(4,3),(8,4) (1,5),(5,6),(9,7) (3,8),(7,9) (0,7),(4,8),(8,9) (3,5),(7,6) (1,2),(5,3),(9,4) (0,5),(4,6),(8,7) (2,8),(6,9) (2,0),(6,1) (3,0),(7,1) Figure 4: The (4,10)-Hoey-Sloane graph with cycle images of GCin bold red. INTEGERS: 25 (2025) 8 3. The Hoey-Sloane Multigraph We begin by stating and proving a result which shows that an edge label (d1, d2) cannot appear on two or more distinct edges of Γ. Theorem 5. If c1, c2,ˆc1,and ˆc2are states on Γ,and (d1, d2)is an input associated with the transitions (c1, c2)and (ˆc1,ˆc2),then (c1, c2) = (ˆc1,ˆc2). Proof. By Equation (1), we have both bc2−c1=nd2−d1and bˆc2−ˆc1=nd2−d1. Reducing both equations modulo b, we have c1≡d1−nd2≡ˆc1(mod b).Since c1 and ˆc1are less than nby definition, it follows that c1= ˆc1.A routine calculation then shows that c2= ˆc2. We now show that any edge (d1, d2) of Minduces some transition (c1, c2) on Γ. Theorem 6. Let (d1, d2)be an edge of M. Then, there are integers 0≤c1≤n−1 and 0≤c2≤n−1such that bc2−c1=nd2−d1. Proof. From our assumption, we know that λ(d1+ (b−n)d2)≤n−1.Then, d1+ (b−n)d2≡d1−nd2≡c1(mod b),where 0 ≤c1≤n−1.It follows that nd2−d1≡ −c1(mod b).We then have, for some integer c2,that nd2−d1=bc2−c1.In another form, bc2=nd2−d1+c1,from which we may say that −b−1≤bc2≤ n(b−1) + n−1=nb −1.Thus, 0 ≤c2≤n−1,and the proof is complete. With Theorems 5 and 6, we may now define a directed, labeled multigraph which we will call the (n, b)-Hoey-Sloane multigraph. Definition 6. Let Nbe the collection of non-negative integers less than n. We map each edge (d1, d2)ofMto the multi-edge (c1, c2)inN×N, which uniquely satisfies Equation (1), to form a labeled multi-edge, which we may visualize in the same fashion as Figure 3. Taking Nas the collection of vertices, along with the collection of labeled multi-edges defined above, we will call this construction the (n, b)-HoeySloane multigraph. To distinguish the multigraph from the usual (n, b)-Hoey-Sloane graph Γ,we will denote the (n, b)-Hoey-Sloane multigraph as ∆. We note that ∆ is simply an alternative representation of Γ,where each multiedge has a unique element from Mas a label, rather than a single edge with a collection of inputs from M. For this reason, when referencing ∆,we will retain the finite-state-machine terminology used when referencing Γ. We now provide two simple examples of the above definition. Example 3. The (2,4)-mother graph Mis seen in Figure 5. Mapping each edge of Mto its multi-edge uniquely determined by Equation (1), we obtain the (2,4)- Hoey-Sloane multigraph ∆ in Figure 6. INTEGERS: 25 (2025) 9 0 1 2 3 Figure 5: The (2,4)-mother graph. 0 start 1 (2,3) (0,2) (1,0) (3,1) (0,0) (2,1) (3,3) (1,2) Figure 6: The (2,4)-Hoey-Sloane multigraph. Example 4. The (3,4)-mother graph Mis displayed in Figure 7. Mapping each edge of Mto its multi-edge, we obtain the (3,4)-Hoey-Sloane multigraph ∆ shown in Figure 8. 0 1 2 3 Figure 7: The (3,4)-mother graph. INTEGERS: 25 (2025) 16 As shown in Example 2, the multiset union C0⊎C1⊎C1⊎C2⊎C2={(9,9),(8,2),(8,2),(2,8),(2,8),(7,1),(7,1),(1,7),(1,7)} may be ordered into permutiple strings. We verify this by examining the multiimage union ∆0⊎∆1⊎∆1⊎∆2⊎∆2corresponding to the above multiset union of cycles, shown in Figure 12. 0 start 3 (2,8) (2,8) (7,1) (7,1) (8,2) (8,2) (1,7) (1,7) (9,9) Figure 12: The multi-image union ∆I= ∆0⊎∆1⊎∆1⊎∆2⊎∆2corresponding to the multiset union CI=C0⊎C1⊎C1⊎C2⊎C2of mother-graph cycles. Since the multi-image union is L-Eulerian, we see, by Corollary 2, that we may order the multiset union C0⊎C1⊎C1⊎C2⊎C2into permutiple strings by traversing Eulerian circuits beginning and ending with the zero state on ∆0⊎∆1⊎∆1⊎∆2⊎∆2. Two examples of these can be found in Example 2. On the other hand, as claimed both in Example 2 and in [11], the multiset union C1⊎C1⊎C2cannot be ordered into permutiple strings. Again, we examine the corresponding multigraph union of multi-images ∆1⊎∆1⊎∆2. 0 start 3 (2,8) (2,8) (7,1) (8,2) (8,2) (1,7) Figure 13: The multi-image union ∆I= ∆1⊎∆1⊎∆2corresponding to the multiset union CI=C1⊎C1⊎C2of mother-graph cycles. Although ∆1⊎∆1⊎∆2contains the zero state and is strongly connected, we see that the indegrees and outdegrees at both vertices are unequal. Thus, the multigraph INTEGERS: 25 (2025) 17 is not L-Eulerian, and the conditions of Corollary 2 are not met. Consequently, C1⊎C1⊎C2cannot be ordered into a permutiple string. 5. Summary, Conclusions, and Future Work In [11], the question is raised regarding sufficient conditions which allow for the formation of permutiple strings. In this paper, we have provided such a condition, which leads to the following equivalence: a multiset union of mother-graph cycles can be ordered into a permutiple string if and only if the multigraph union of corresponding cycle multi-images is L-Eulerian (that is, if it contains the zero state, is strongly connected, and the indegree is equal to the outdegree at each vertex). From the above considerations, we see that counting permutiples with a fixed base, multiplier, and length, and a fixed multiset of digits, is reduced to counting Eulerian circuits of a particular strongly-connected multi-image union containing the zero state having equal indegree and outdegree at each vertex. Counting Eulerian circuits of directed graphs may be accomplished in polynomial time using the BEST algorithm [17]. Multigraph variations also exist [3]. That said, if we lift the constraint of a fixed multiset of digits, then counting permutiples of a fixed base, multiplier, and length ℓ, becomes a problem of finding all multi-image unions which are L-Eulerian and have exactly ℓmulti-edges. We are uncertain of the difficulty of this latter problem, and it presents an area for further study. Adding to the difficulties presented above, for larger bases and multipliers, the number of cycle multi-images increases rapidly. For example, as stated in [10], there are 986 cycles of the (4,10)-mother graph. Thus, considering (4,10)-permutiples of a fixed length ℓ, there are a very large number of possible cycle multi-image unions to consider. Filtering these possibilities down to those which are L-Eulerian appears to be a difficult task, and we leave it to future efforts. As mentioned in [10], the methods presented here may be applied to the more specific palintiple problem where σis the reversal permutation. Finding palintiple numbers means that we only need to examine the cycle multi-images of mothergraph 1and 2-cycles. However, finding palintiple strings, which have the very particular form (d0, dk)(d1, dk−1)···(dj, dk−j)···(dk−j, dj)···(dk−1, d1)(dk, d0), seems to be more difficult than one would expect; these are not obvious when examining the Hoey-Sloane graph or the individual cycle multi-images. This is yet another open question which could advance our understanding of palintiple numbers. In contrast to the above, for the very specific case of palintiple numbers when n+1 divides b, Sloane [16] shows that the number of k+1-digit (n, b)-palintiples, for k≥3, INTEGERS: 25 (2025) 18 is F⌊k+1 2⌋−1,where Fmis the mth Fibonacci number. Restating this result in terms of the permutiple class Cconsidered in Example 7, there are F⌊k+1 2⌋−1palintiple strings of length k+ 1.Other significant integer sequences which might arise when considering the general permutiple problem presents another open invitation for further investigation. We conclude by presenting some questions relating to the “derived-permutiple” problem mentioned by [8, 9, 10]. A base-bpermutiple (dk, . . . , d0)bis derived if its carries cj, not including the trivial carry c0= 0,are the digits of a base-npermutiple (ck, ck−1, . . . , c1)n.This phenomenon is illustrated by the (6,12)-permutiple (10,3,5,1,8,6)12 = 6 ·(1,8,6,10,3,5)12 whose non-trivial carries are the digits of the (2,6)-palintiple (4,3,5,1,2)6= 2 ·(2,1,5,3,4)6.If we allow for leading digits which are zero, we have another pair of examples related to one another: considering the (2,3)-palintiple (2,1,0,1)3= 2 ·(1,0,1,2)3,the (3,6)-permutiple (2,1,0,4,3)6= 3 ·(0,4,2,1,3)6has the carry vector (2,1,0,1,0),and the (6,12)- permutiple (8,1,6,4,3,0)12 = 6·(1,4,3,0,8,6)12 has the carry vector (2,1,0,4,3,0), whose non-trivial entries are the digits of the previous example. From the multigraph perspective developed here, the derived-permutiple problem may be posed as the following: given a base-npermutiple (ck, ck−1, . . . , c1)n,find an (n, b)-HoeySloane multigraph for which the circuit (0, c1, . . . , ck−1, ck,0) may be traversed by using a collection of inputs which is a multiset union of (n, b)-mother-graph cycles. The above examples naturally lead to several questions: Given (dk, . . . , d0)b,derived from (ck, ck−1, . . . , c1)n,under what conditions can we find a (b,ˆ b)-permutiple derived from (dk, . . . , d0)b? Are there cases for which this process may be continued indefinitely? Can entire permutiple classes (see Definition 3) be constructed from others? What larger patterns might exist between these classes? Other questions regarding the general problem may be found in [8, 9]. Acknowledgement. We are grateful to the anonymous referee for their time, as well as their valued suggestions and corrections which greatly enhanced the quality of this work. References [1] J. Bang-Jensen and G. Z. Gutin, Digraphs: Theory, Algorithms and Applications, Springer, London, 2009. [2] X. Faber and J. Grantham, On integers whose sum is the reverse of their product, Fibonacci Quart. 61(1) (2023), 28-41. [3] M. Farrell and L. Levine, Multi-Eulerian tours of directed graphs, Electron. J. Combin. 23(2) (2016), P2.21. [4] S. Guttman, On cyclic numbers, Amer. Math. Monthly 41(3) (1934), 159-166. INTEGERS: 25 (2025) 19 [5] D. J. Hoey, Untitled online article (not peer-reviewed). Available from OEIS Foundation Inc. (1992), The On-Line Encyclopedia of Integer Sequences at https://oeis.org/A008919/ a008919.txt. [6] B. V. Holt, Some general results and open questions on palintiple numbers, Integers 14 (2014), #A42. [7] B. V. Holt, Derived palintiple families and their palinomials, Integers 16 (2016), #A27. [8] B. V. Holt, On permutiples having a fixed set of digits, Integers 17 (2017), #A20. [9] B. V. Holt, A method for finding all permutiples with a fixed set of digits from a single known example, Integers 24 (2024), #A88. [10] B. V. Holt, Finding permutiples of a known base and multiplier, Integers 25 (2025), #A13. [11] B. V. Holt, Utilizing the reflective symmetry of the mother graph in finding new permutiple classes from old, preprint, arXiv:2504.17158v2. [12] D. Kalman, Fractions with cycling digit patterns, College Math. J. 27(2) (1996), 109-115. [13] L. H. Kendrick, Young graphs: 1089 et al., J. Integer Seq. 18 (2015), Article 15.9.7. [14] B. Qu and S. J. Curran, An unexpected digit permutation from multiplying in any number base, in Combinatorics, Graph Theory and Computing, Springer, Cham, 2022, 13-31. [15] D. Silberger, Eulerian directed multigraphs, preprint, arXiv:2408.12699. [16] N. J. A. Sloane, 2178 and all that, Fibonacci Quart. 52(2) (2014), 99-120. [17] T. van Aardenne-Ehrenfest and N. G. de Bruijn, Circuits and trees in oriented linear graphs, Bull. Belg. Math. Soc. Simon Stevin 28 (1951), 203-217. [18] R. Webster and G. Williams, On the trail of reverse divisors: 1089 and all that follow, Math. Spectrum 45 (2012/2013): 96-102. [19] A. L. Young, k-reverse multiples, Fibonacci Quart. 30 (1992), 126-132. [20] A. L. Young, Trees for k-reverse multiples, Fibonacci Quart.,30 (1992), 166-174.