The 1-2-3 Conjecture, posed in 2004 by Karonski, Luczak, and Thomason, is as follows:" If G
is a graph with no connected component having exactly 2 vertices, then the edges of G may …

For a connected graph G of order| V (G)|≥ 3 and a k-labelling c: E (G)→{1, 2,…, k} of the
edges of G, the code of a vertex v of G is the ordered k-tuple (ℓ1, ℓ2,…, ℓk), where ℓi is the …

For any S⊂ ℤ we say that a graph G has the S-property if there exists an S-edge-weighting
w: E (G)→ S such that for any pair of adjacent vertices u, v we have Σ_ {e∈ E (v)} w (e)≠ Σ …

In 2004, Karoński, Łuczak and Thomason proposed 1-2-3-Conjecture: For every nice graph
G there is an edge weighting function w: E (G)→{1, 2, 3} such that the induced vertex …

For a connected graph G of order| V (G)|≥ 3 and a k-edge-weighting c: E (G)→{1, 2,…, k} of
the edges of G, the code, codec (v), of a vertex v of G is the ordered k-tuple (ℓ 1, ℓ 2,…, ℓ k) …

A total k-coloring of a graph G is a coloring of vertices and edges of G using colors of the set
{1,..., k}. These colors can be used to distinguish adjacent vertices of G. There are many …

Abstract The well-known 1-2-3 Conjecture asks whether almost all graphs can have their
edges labelled with 1, 2, 3 so that any two adjacent vertices are distinguished wrt the sums …

In this thesis we explore graph decompositions under different constraints. The title of the is
due to the fact that most of these decompositions are neighbour-distinguishing. That is, we …

An assignment of numbers to the vertices of graph G is closed distinguishing if for any two
adjacent vertices v and u the sum of labels of the vertices in the closed neighborhood of the …

Abstract The 1-2-3 Conjecture, posed by Karonski, Luzcak, and Thomason in 2004, states
that any connected graph on at least three vertices can be edge-weighted from {1, 2, 3} so …