The Observability of some regular graphs

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The Observability of some regular graphs

• Janka Rudaová
• Institute of Mathematics
• Faculty of Science
• P. J. afárik University, Koice
• Budmerice 2005

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Observability of graph

• Let G be a simple graph. An edge coloring f 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.

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• 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).

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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)
8, 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 ?s(G)), and Cerný, Hornák and
Soták (obs(G)).

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Lower bound
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Known results
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• Open questions
• case of regular graphs of low degree
• arbitrary number of copies of some graphs
• ...

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Some 3-regular graphs
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Theorem Let k be a minimal integer for which
6p is valid. Than obs( pK3,3) ? k 1. Idea
of proof proof is by induction on the p 1.
p1, k 5 2.
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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.
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Theorem Let G be 3-regular graph at the most of
8 vertices. Let k be the smallest integer for
which the following pV(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.
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• The modifications of the coloring
• If the regularity is not needed then we get ?0(G)
  the vertex-distinguishing index (Harary,
  Plantholt 1985).
• If the regularity is needed but we want to
  distinguish only nieghbour vertices so we get
  ?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.

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Thanks for your attention.