The edge total irregularity strength of complete graphs

1. The edge total irregularity strength of complete graphs Roman Soták (joint work with S. Jendroľ and J. Miškuf) Institute of Mathematics Faculty of Science P. J. Šafárik University, Košice Budmerice 2005

2. The weight of the edge xy under a total labeling δ: VE →{1,2,…,k} is wt(xy) = δ(x) + δ(xy) + δ(y). For a graph G = (V,E) we define a labeling δ: VE →{1,2,…,k} to be an edge irregular total k-labeling of the graph G if wt(e) ≠ wt(f) for every two distinct edges e and f. The minimum k for which the graph G has an edge irregular total k-labeling is called the edge total irregularity strength of the graph G, tes(G). Theorem (M. Bača, S. Jendroľ, M. Miller, J. Ryan ‘01): tes(G) ≥ max {(|E|+2)/3, (Δ+1)/2} Budmerice 2005

3. tes(K5) • lower bound: 4 • if tes(K5) = 4, then weights of edges form {3,4, …,12} 1 4 1 3 ? 1 4 2 4 1 4 • used weights: 3, 6, 7, 8, 9, 12 • label of last vertex: if 1 or 2 then weight 11 is not attainable, if 3 or 4 then weight 4 is not attainable • tes(G) ≥ 5 • tes(G) = 5 by labeling last vertex by 2 and incident edges by 1,2,4,5. Budmerice 2005

4. Trees Theorem (J. Ivančo, S. Jendroľ ’05): Let T be an arbitrary tree. Then Note: Moreover there exists a labeling with δ(x)  {1, tes(T)} for each vertex x. Budmerice 2005

5. Complete graphs Theorem: For n > 1, n ≠ 5 • Sketch of the proof: • t := (|E|+2)/3 • we label (n+1)/3 vertices by 1 and (n+1)/3 vertices by t • then for numbers of edges: |E11| + |E1*| = |Ett| + |Et*| = t – 1; |E1t| + |E**|  {t – 1, t} V1 E11 E1* V* E1t E** Et* Ett Vt Budmerice 2005

6. Complete bipartite graphs Theorem: For 1 < n  m • Sketch of the proof: • t := (|E|+2)/3 • we label (n+1)/3 vertices of the first part by 1 and (n+1)/3 vertices by t, (m+1)/3 vertices of the second part by 1 and (m+1)/3 vertices by t • then |E11| + |E1*| = |Ett| + |Et*| = t–1; |E1t| + |E**|  {t–2, t–1, t} Budmerice 2005

7. Thanks for your attention. Budmerice 2005