Graceful polynomials of small degree

Graceful polynomials of small degree Andrea Vietri 1 1 Dipartimento di Scienze di Base e Applicate per l Ingegneria Sapienza Università di Roma WAGTCN2018, Naples, 13-14 September 2018 Andrea Vietri 1 / 25

Outline 1 Basic concepts 2 Graceful polynomials of small degree (vanishing (mod 2) ) 3 Final remarks Andrea Vietri 2 / 25

Basic concepts Why graceful polynomials? For any graph we define a family {S n } n N of graceful polynomials; S n has degree n. Andrea Vietri 3 / 25

Basic concepts Why graceful polynomials? For any graph we define a family {S n } n N of graceful polynomials; S n has degree n. There is a deep connection between these polynomials and graceful labellings. The prototype, S 1, was employed by A. Rosa (1967) to obtain the most important class of non-graceful graphs. Andrea Vietri 3 / 25

Basic concepts Graceful labellings A graph G = (V, E) is graceful if f : V {0, 1, 2,..., E } injective and such that { f (u) f (v) : uv E} = {1, 2,..., E }. If a graph admits no such labelling, it is non-graceful. 8 8 0 4 4 6 0 6 7 3 1 2 5 7 1 2 3 4 1 Non-graceful Andrea Vietri 4 / 25

Basic concepts Example of graceful polynomial How to obtain a graceful polynomial from G = : Andrea Vietri 5 / 25

Basic concepts Example of graceful polynomial How to obtain a graceful polynomial from G = : Assign a variable to each vertex: x 1 x 4 x 2 x 3 x 5 x 6 x 7 Andrea Vietri 5 / 25

Basic concepts Example of graceful polynomial How to obtain a graceful polynomial from G = : x 1 Assign a variable to each vertex: x 4 Fix a positive integer n, say n = 3... x 2 x 3 x 5 x 6 x 7 Andrea Vietri 5 / 25

Basic concepts x 1 x 2 Take any set of 3 edges and store : x 5 x 6 x 7 (x 1 x 2 )(x 2 x 5 )(x 6 x 7 ) x 1 x 2 x 6 + x 1 x 2 x 7 +... (mod 2) Andrea Vietri 6 / 25

Basic concepts x 1 x 2 Take any set of 3 edges and store : x 5 x 6 x 7 (x 1 x 2 )(x 2 x 5 )(x 6 x 7 ) x 1 x 2 x 6 + x 1 x 2 x 7 +... (mod 2) SUM UP these polynomials over all the ( 8 3) sets of 3 edges: this is S 3 (G), the 3-rd graceful polynomial of G. Andrea Vietri 6 / 25

Basic concepts Formally... given a graph G... Assign variable x i to vertex v i. Andrea Vietri 7 / 25

Basic concepts Formally... given a graph G... Assign variable x i to vertex v i. Associate any given edge e j = v p v q to the polynomial P j = x p + x q. Andrea Vietri 7 / 25

Basic concepts Formally... given a graph G... Assign variable x i to vertex v i. Associate any given edge e j = v p v q to the polynomial P j = x p + x q. The n-th graceful polynomial of G is SG n (x 1, x 2,..., x V ) P j1 P j2 P jn (mod 2). 1 j 1 <j 2 <...<j n E Andrea Vietri 7 / 25

Basic concepts Graceful polynomials and non-graceful graphs The following Lemma provides the connection: Lemma (A.V. 2012) Let G be a graceful graph and let f i be the label of vertex v i. Then, for every positive integer n E(G), ( [ E +1 ) SG n (f 1,..., f V ) 2 ] (mod 2). n Therefore, if some graceful polynomial VANISHES (mod 2)... Andrea Vietri 8 / 25

Main problem: Choose n, and characterise all graphs for which SG n vanishes (mod 2). Andrea Vietri 9 / 25

Main problem: Choose n, and characterise all graphs for which SG n vanishes (mod 2). The problem seems interesting on its own right! Andrea Vietri 9 / 25

Main problem: Choose n, and characterise all graphs for which SG n vanishes (mod 2). The problem seems interesting on its own right! However, once we have these graphs, we can also find non-graceful graphs (choosing a suitable number of edges, if we can...). Andrea Vietri 9 / 25

Main problem, degree 1 This is Rosa s polynomial. (A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs, Internat. Sympos. Rome, 1966, pp. 349-355) Andrea Vietri 10 / 25

Main problem, degree 1 This is Rosa s polynomial. (A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs, Internat. Sympos. Rome, 1966, pp. 349-355) Let δ p denote the degree of vertex v p. SG 1 (x 1, x 2,..., x V ) P j (x p + x q ) δ p x p 1 j E v pv q edge v p vertex Andrea Vietri 10 / 25

δp x p vanishes if all degrees are even (Eulerian graph). Andrea Vietri 11 / 25

δp x p vanishes if all degrees are even (Eulerian graph). On the other hand, ( [ E +1 ]) 2 is odd if E 1, 2 (mod 4). Therefore: 1 Andrea Vietri 11 / 25

δp x p vanishes if all degrees are even (Eulerian graph). On the other hand, ( [ E +1 ]) 2 is odd if E 1, 2 (mod 4). Therefore: 1 Theorem (Rosa) Eulerian graphs with E 1, 2 (mod 4) are non-graceful. Andrea Vietri 11 / 25

Degree 2 Denote coefficients of graceful polynomials by H. S 2 G H 2 px 2 p + H 11 pqx p x q. Andrea Vietri 12 / 25

Degree 2 Denote coefficients of graceful polynomials by H. S 2 G H 2 px 2 p + H 11 pqx p x q. H 2 p counts the PAIRS of edges containing v p : v p (x p + x a )(x p + x b ) v b = xp 2 +... v a Andrea Vietri 12 / 25

Degree 2 Denote coefficients of graceful polynomials by H. S 2 G H 2 px 2 p + H 11 pqx p x q. H 2 p counts the PAIRS of edges containing v p : Therefore, H 2 p ( δ p 2 ) (mod 2) v p (x p + x a )(x p + x b ) v b = xp 2 +... v a δ p 0, 1 (mod 4) v p if we ask for vanishing. Andrea Vietri 12 / 25

As to H 11 pq, there are two cases. G 2 1 v p v q δ p δ q choices Andrea Vietri 13 / 25

As to H 11 pq, there are two cases. G 2 1 G 2 v 2 p v q v p v q δ p δ q choices δ p δ q 1 choices: (x p + x q )(x p + x q ) not allowed Andrea Vietri 13 / 25

As to H 11 pq, there are two cases. G 2 1 G 2 v 2 p v q v p v q δ p δ q choices δ p δ q 1 choices: So we have (if we ask for vanishing): (x p + x q )(x p + x q ) not allowed δ p 0, 1 v p, (δ p, δ q ) (1, 1) G 2 1, (δ p, δ q ) (1, 1) G 2 2 all (mod 4) Andrea Vietri 13 / 25

From the above conditions we obtain all possible graphs: Theorem (A.V. 2012) SG 2 vanishes for complete graphs on 4d + 2 vertices, for any integer d. No other graph satisfies the requirement. By extending Rosa s counting technique to this level we obtain: Theorem (A.V. 2012, new proof of an old result! ) Complete graphs on either 16u + 10 or 16u + 14 vertices, with any integer u, are non-graceful. (count the edges and reach a contradiction through the above Lemma) Andrea Vietri 14 / 25

Degree 3 S 3 G H 3 px 3 p + H 21 p,qx 2 p x q + H 111 pqr x p x q x r (commas separate SETS of indices) Andrea Vietri 15 / 25

Degree 3 S 3 G H 3 px 3 p + H 21 p,qx 2 p x q + H 111 pqr x p x q x r (commas separate SETS of indices) H 3 p ( δ p 3 ) Andrea Vietri 15 / 25

Degree 3 S 3 G H 3 px 3 p + H 21 p,qx 2 p x q + H 111 pqr x p x q x r (commas separate SETS of indices) H 3 p ( δ p 3 ) Hp,q(G 21 1 2) ( δ p ) δq, H 21 2 p,q(g2 2) ( δ p ) δq (δ 2 p 1) G 2 v a 1 G 2 v 2 p v q v p v q (x p + x q ) 2 (x p + x a ) not allowed Andrea Vietri 15 / 25

As to H 111 pqr, there are 4 subgraphs on 3 vertices: v p v p v p v p v q v r v q v r v q v r v q v r G 3 1 G 3 2 G 3 3 G 3 4 Andrea Vietri 16 / 25

As to H 111 pqr, there are 4 subgraphs on 3 vertices: v p v p v p v p v q v r v q v r v q v r v q v r G 3 1 G 3 2 G 3 3 G 3 4 For example, H 111 pqr (G 3 3 ) δ pδ q δ r δ r δ q. (forbidden repetitions of either v p v q or v p v r ) v q v p v r Andrea Vietri 16 / 25

System of constraints: v p δ p 3 (mod 4) G1 2 ( δp ) δq + ( δ q ) δp 0 (mod 2) 2 2 G 2 ( δp ) 2 δq + ( δ q ) δp + δ 2 2 p + δ q 0 (mod 2) (Hp,q 21 + Hq,p 21...) G1 3 δ p δ q δ r 0... G2 3 δ p (δ q δ r + 1) 0 G3 3 δ p δ q δ r + δ q + δ r 0 G4 3 δ p δ q δ r + δ p + δ q + δ r 0 Andrea Vietri 17 / 25

If the graph has some vertices of odd degree: Andrea Vietri 18 / 25

If the graph has some vertices of odd degree: Theorem (A.V. 2016) S 3 G vanishes (mod 2) on G precisely in one of the following two cases. (A) G is a complete graph K 4t+2, for some positive integer t, possibly having 4u additional vertices and 4u(4t + 2) additional edges that connect these vertices to the above complete graph. (B) G is obtained by taking two complete graphs K a, K b and n additional vertices satisfying one of the following conditions: (B 1 ) : a b 2 (mod 4) and n = 0; (B 2 ) : a b n 1 (mod 4) except a = b = n = 1; (B 3 ) : a b n 3 (mod 4); every additional vertex must be fully connected to K a and K b. Andrea Vietri 18 / 25

Old and new non-graceful graphs, as a consequence: Theorem The following graphs are non-graceful. Type (A) in the above theorem, provided t 2 (mod 4). Type (B 1 ) with a + b 12 (mod 16); type (B 2 ) with a + b 10 (mod 16); type (B 3 ) with a + b 14 (mod 16). K 7 K 7 K 5 K 5 a = b = 7, n = 3 a = b = n = 5 Andrea Vietri 19 / 25

Degree 4 Not surprising: much more cases and many subgraphs to check... Andrea Vietri 20 / 25

Degree 4 Not surprising: much more cases and many subgraphs to check... S 4 G p H 4 px 4 p + p,q H 31 p,qx 3 p x q + pq H 22 pqx 2 p x 2 q + + p,qr Hp,qr 211 xp 2 x q x r + Hpqrs 1111 x p x q x r x s. pqrs Andrea Vietri 20 / 25

Degree 4 Not surprising: much more cases and many subgraphs to check... S 4 G p H 4 px 4 p + p,q H 31 p,qx 3 p x q + pq H 22 pqx 2 p x 2 q + + p,qr Hp,qr 211 xp 2 x q x r + Hpqrs 1111 x p x q x r x s. pqrs Strategy: (I) Find constraints for single vertices and pairs of vertices (II) Constraints for subgraphs with 3 vertices some ruled out by (I) Andrea Vietri 20 / 25

Some constraints for the degree 4 Constraint for single vertices: 0 δ p 3 (mod 8) Andrea Vietri 21 / 25

Some constraints for the degree 4 Constraint for single vertices: 0 δ p 3 (mod 8) Forbidden degrees of non-adjacent vertices: (1, 3), (2, 2), (2, 3), (3, 3) (mod 4) Andrea Vietri 21 / 25

Some constraints for the degree 4 Constraint for single vertices: 0 δ p 3 (mod 8) Forbidden degrees of non-adjacent vertices: (1, 3), (2, 2), (2, 3), (3, 3) (mod 4) Forbidden degrees of adjacent vertices: (0, 0), (0, 1), (3, 3) (mod 4) In particular, there is at most one vertex of degree 3 (mod 4) Andrea Vietri 21 / 25

Degree 4: in the end... Only two very small graphs have passed ALL the exams! Andrea Vietri 22 / 25

Degree 4: in the end... Theorem Only two very small graphs have passed ALL the exams! Let G be a graph having at least 4 edges and some odd vertices. The polynomial SG 4 vanishes (mod 2) on G if and only if G is a 3-cycle together with a further edge which can be either pendent or disjoint from the cycle. Andrea Vietri 22 / 25

Final remarks Final remarks Degree 5 is being analysed... Andrea Vietri 23 / 25

Final remarks Final remarks Degree 5 is being analysed... Are there efficient techniques for higher degrees? Andrea Vietri 23 / 25

Final remarks Final remarks Degree 5 is being analysed... Are there efficient techniques for higher degrees? Why only vanishing (mod 2)? Andrea Vietri 23 / 25

Final remarks Final remarks Degree 5 is being analysed... Are there efficient techniques for higher degrees? Why only vanishing (mod 2)? We have a FAMILY OF POLYNOMIALS, let s study it! Andrea Vietri 23 / 25

For Further Reading Final remarks J.A. Gallian A Dynamic Survey of Graph Labeling Electr. J. Comb. 16, DS6 (2015). A. Rosa On certain valuations of the vertices of a graph Theory of Graphs (Internat. Sympos. Rome, 1966) pp. 349-355, Gordon and Breach, New York; Dunod, Paris, 1967. A. Vietri Necessary conditions on graceful labels: a study case on trees and other examples Util. Math. 89 (2012), pp. 275-287. A. Vietri On the Graceful Polynomials of a graph submitted... Andrea Vietri 24 / 25

Final remarks THANK YOU. Andrea Vietri 25 / 25