The Observability of some regular graphs

1. The Observability of some regular graphs Janka Rudašová Institute of Mathematics Faculty of Science P.J.Šafárik University, Košice Budmerice 2005

2. Observability of graph Let G be a simple graph. An edge coloringf of a graph G is an assignment of colors from the set {1, 2, …, k} to the edges of G. We denote nd the number of the vertices of degree d. For vertex v we denote by Ff (v) the set of colors assigned to the set of edges incident to v. We can say that the colouring is regular, if no two adjacent edges have the same color. We call coloring the vertex-distinguishing, if Ff (u) Ff (v) for any two distinct vertices u, v.

3. 123 246 123 235 156 345 134 345 • if the graph consists of two isolated vertices or an isolated edge, then for this graph does not exist vertex-distinguishing coloring. To the contrary, there is always at least one regular vertex distinguishing coloring (eg. coloring binding different color to each edge).

4. For the graphs which do not consist of two isolated vertices nor isolated edge the observability is defined as a minimal k for which there are regular vertex-distinguishing coloring, using exactly k colors. The observability of graph G is represented as obs(G). (obs(G) = ∞, if for the graph G does not exist the vertex- distinguishing coloring.) Indpendently of each other, this notion was established by Burris and Schelp (vdi(G) or χs’(G)), and Černý, Horňák and Soták (obs(G)).

5. Hypothesis: Let G be a graph for which there are vertex- distinguishing coloring. Let k is a minimal number for which the following statement is valid: ≥ nd for each dZ : δ(G)  d  Δ(G). Then k obs(G)  k + 1. For all graphs G for those which the value obs(G) is known, is the hypothesis valid (eg. Kn, Cn, Pn, Wn, Km,n,...). Lower bound

6. Balister, Bollabas and Schelp: Let G be the vertex disjoint unionof paths in the lenght at least 2. Let k be a minimal integer for which the following n1  k and is valid. Then obs(G)  k + 1. • Balister: 2- regular graphs • Simple 2 – regular graph of order n is observable by k colors if and only if • k is odd and either or • or • k is even and Known results

7. Balister, Kostochka, Li a Schelp – they have shown the validity of the hypothesis for all the graphs G, for which the following statement is valid: and δ(G) ≥ 5. • Open questions: • case of regular graphs of low degree • arbitrary number of copies of some graphs • ...

8. n 4 2 • Taczuk and Woźniak: two classes of 3- regular graphs - • if k is a minimal integer for which ≥ n is valid, then • obs( Ln)  k + 1 • if k is minimal integer for which ≥ 4p is valid, than • obs( pK4)  k + 1 1 3 n-1 Some 3-regular graphs

9. 123 125 135 134 245 235 p . . . . . . p1 p2 Theorem :Let kbe a minimal integer for which ≥ 6p is valid. Than obs( pK3,3)  k + 1. Idea of proof : proof is by induction on thep 1. p=1, k =5 2.

10. 123 126 125 236 135 136 134 156 245 456 235 146 Notice:obs(2K3,3) = 6, obs(3K3,3) = 7 If obs(3K3,3) = 6 so there are not used exactly 2 different 3 elements color sets. Each color is in 10 color sets, so that if is in unused set so it is shown only on 4 edges in the graph and it is represented in both missing sets. So the missing sets are equal.

11. Theorem: Let G be 3-regular graph at the most of 8 vertices. Let k be the smallest integer for which the following ≥ p|V(G)| is valid. Than obs( pG)  k + 1. Theorem: In addition if 3-regular graphs consist of just components to maximal 8 vertices the mentioned hypothesis is valid again. Theorem: The hypothesis is valid for nK4,4, nK5,5, nK6,6, nK7,7, nK6.

12. The modifications of the coloring • If the regularity is not needed then we get χ0(G) – the vertex-distinguishing index (Harary, Plantholt 1985). • If the regularity is needed but we want to distinguish only nieghbour vertices so we get χ’a(G) – the adjacent vertex-distinguishing proper edge coloring number (Zhongfu, Liu, Wang 2002). • If we suppose the coloring of the vertices and we want to distinguish the edges so we get next invariants.

13. Thanks for your attention.