1 2 3 4 5 6 7 8 The Indifference Graph Numerical Monoid 9 10 Conjecture 11 12 K. Hilmer and V. Ponomarenko 13 14 July 28, 2021 15 16 17 Abstract - A numerical monoid is a subset of the nonnegative integers, containing zero, 18 that is closed under addition. We conjecture that every indifference graph whose vertex 19 degrees all appear in a numerical monoid, must have its order appear in that same numerical monoid. We offer several results in the direction of this conjecture. 20 21 Keywords : indifference graph; interval graph; intersection graph; numerical monoid; numerical semigroup 22 Mathematics Subject Classification (2020) : 05C10 23 24 25 1 Introduction 26 An indifference graph is an undirected graph whose vertices are a finite (multi)set of real 27 numbers, and whose edges are those pairs of vertices which are within distance one of 28 each other, as real numbers. Since each vertex is within one of itself, we take a loop at 29 each vertex. Indifference graphs are an important class of graphs with many applications 30 such as to order theory and algebra, and are the subject of considerable study (see, e.g., 31 [3, 4, 5, 6, 9, 10, 12, 14]). 32 p 33 2 2 e 3 π 34 35 Figure 1: An indifference graph 36 A numerical monoid is a subset of the nonnegative integers , containing 0, closed 37 N0 under addition. A numerical monoid that is also cofinite is called a numerical semigroup. 38 Both of these are important objects (particularly numerical semigroups), themselves well- 39 studied (see, e.g., [1, 2]). 40 For graph G, we denote by jGj the order of jGj, i.e. the number of vertices of G. For 41 vertex v 2 G, we denote by deg(v) the degree of v, i.e. the number of edges incident to 42 v. We consider a loop at v to contribute 1 to deg(v); hence the example in Figure 1 has 43 deg(π) = 3. Given a graph G and a numerical monoid S, we say that G respects S if: 44 (8v 2 G; deg(v) 2 S) ! jGj 2 S 45 46 the pump journal of undergraduate research 0 (2017), 000{000 1 47 48 49 50 1 2 3 4 5 Note that G can respect S vacuously, if the degree of some vertex of G does not 6 appear in S. If instead all vertex degrees of G appear in S, then the order of G must 7 also appear in S, for G to respect S. By a; b we mean the integers between a and b, 8 inclusive; similarly, by a; 1 we mean allJ integersK greater than or equal to a. For any 9 J M set S ⊆ N0, by kS we mean fkn : n 2 Sg. Note that the example in Figure 1 respects 10 2N0 (vacuously), respects 2; 1 and 2; 5 , but does not respect 2; 4 . 11 Inspired by a recent resultJ (TheoremM J 2.1K below), we propose severalJ K conjectures. Al- 12 though we are not able to prove any of them, we do have a variety of partial results. 13 Conjecture 1.1 Every indifference graph respects every numerical monoid. 14 15 Conjecture 1.2 Every indifference graph respects every numerical semigroup. 16 17 Conjecture 1.3 Every indifference graph respects kN0, for all k 2 N0. 18 Because kN0 and numerical semigroups are each numerical monoids, Conjecture 1 19 implies the others. Perhaps surprisingly, Conjecture 2 is equivalent to Conjecture 1. 20 21 Theorem 1.4 If Conjecture 2 holds, then Conjecture 1 holds. 22 Proof. Let G be an indifference graph and S a numerical monoid such that 8v 2 23 G; deg(v) 2 S. Set S0 = S [ jGj + 1; 1 . Note that S0 is a numerical semigroup that 24 agrees with S on 0; jGj . Hence,J 8v 2 G; deg(M v) 2 S0. Applying Conjecture 2, G respects 0 0 J K 25 S , so jGj 2 S . Hence jGj 2 S, so G respects S. 26 In the remainder, we approach these conjectures from several directions. In Section 27 2, we consider fixed numerical monoids/semigroups respected by all indifference graphs. 28 In Section 3, we instead consider fixed indifference graphs that respect every numerical 29 monoid or every kN0. 30 31 2 Fixed Numerical Monoids 32 33 In this section, we consider specific numerical monoids that are respected by many indif- ference graphs. 34 We begin with the motivating theorem for these conjectures. It appears in [8] in 35 contrapositive form. For completeness, we provide a proof. 36 37 Theorem 2.1 (Balof/Pinchasi) Every indifference graph respects 2N0. 38 Proof. Let G be an indifference graph with adjacency matrix M. Suppose that deg(v) is 39 even for every v 2 G. Then every row sum of M is even, hence the sum of all the entries 40 of M is even. Now consider M − I. This is symmetric with 0 along the diagonal, so the 41 sum of all its entries is even. Combining these results, we conclude that the sum of all 42 entries of I (namely, jGj) is even. 43 We collect some basic properties in the following theorem. 44 45 46 the pump journal of undergraduate research 0 (2017), 000{000 2 47 48 49 50 1 2 3 4 5 Theorem 2.2 Let G be a graph, S; S0 be numerical monoids, and a 2 . 6 N0 7 1. Every indifference graph respects 0N0 = f0g. 8 9 2. Every indifference graph respects 1N0 = N0. 10 3. If G respects S, then G also respects S [ a; 1 . 11 J M 0 0 12 4. If G respects S and S , then G also respects S \ S . 13 5. If G respects S, then G also respects S n 1; a + 1 . 14 J K 15 Proof. (1) Every vertex in an indifference graph has degree at least one, since we assume it has a loop. Hence, all nonempty graphs respect 0 vacuously, and the empty graph 16 N0 respects it nonvacuously. 17 (2) clear 18 (3) Suppose all vertices of G have their degree in S[ a; 1 . If any vertex v has deg(v) 2= S, 19 then deg(v) ≥ a; in this case, jGj ≥ a (lookingJ at theM neighbors of v alone), and so 20 jGj 2 S [ a; 1 . Otherwise, all vertices v 2 G have deg(v) 2 S. Since G respects S by 21 hypothesis,J we haveM jGj 2 S ⊆ S [ a; 1 . 22 (4) Suppose all vertices of G have theirJ degreesM in S \ S0. Then, in particular, all vertices 23 of G have their degrees in S. Since G respects S, jGj 2 S. Repeating for S0 we find 0 0 24 jGj 2 S ; hence jGj 2 S \ S . 0 25 (5) By (1) and (3), G respects S = f0g [ a + 2; 1 ; now apply (4). We combine Theorems 2.1 and 2.2 to getJ the followingM result. 26 27 Corollary 2.3 Let t 2 N be odd. Every indifference graph respects the numerical semi- 28 group h2; ti = f2x + ty : x; y 2 N0g. 29 Proof. h2; ti = 2 [ t; 1 . 30 N0 We can move fromJ one numericalM monoid to another with the following result. 31 32 Theorem 2.4 Let S be a numerical monoid and k 2 N0. Suppose that every indifference 33 graph respects kS. Then every indifference graph respects S. 34 Proof. Let G be an indifference graph with deg(v) 2 S for all v 2 G. We produce a new 35 indifference graph G0 which has the same vertices as G, only each repeated k times. For 0 36 every v 2 G, there are k copies v1; v2; : : : ; vk 2 G with k deg(v) = deg(v1) = deg(v2) = 0 0 37 ··· = deg(vk). Hence deg(vi) 2 kS for all vi 2 G . Since G respects kS, we have 0 0 38 jG j 2 kS. Hence there is some n 2 S with kn = jG j = kjGj, so jGj 2 S. 39 In particular, Theorem 2.4 tells us that if a numerical monoid T is respected by all 40 indifference graphs, we can set k = gcd(T ). Then T = kS for a numerical monoid S with 41 gcd(S) = 1, and now all indifference graphs respect S. It is well-known (e.g. [1]) that a numerical monoid with no common factor is a numerical semigroup, so in fact S is a 42 numerical semigroup. 43 For our remaining results, it will be useful to develop some machinery. Let G be 44 an indifference graph with vertices v1 ≤ v2 ≤ · · · ≤ vn. We relabel these real values 45 46 the pump journal of undergraduate research 0 (2017), 000{000 3 47 48 49 50 1 2 3 4 5 by their integer indices 1; 2; : : : n. The neighborhood of each vertex of G will then form 6 some integer interval. For vertex i 2 1; n , define L(i) and R(i) to be the left and right 7 endpoints of that interval. That is, iJ is connectedK to each element of L(i);R(i) , and 8 nothing else. Note that L(i) ≤ i ≤ R(i); hence L(1) = 1 and R(n) = n. WeJ also haveK the 9 key property that deg(i) = R(i)−L(i)+1. By symmetry, j ≤ R(i) if and only if i ≥ L(j).