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    The distinguishing index of infinite graphs

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    The distinguishing index D0(G) of a graph G is the least cardinal d such that G has an edge colouring with d colours that is only preserved by the trivial automorphism. This is similar to the notion of the distinguishing number D(G) of a graph G, which is defined with respect to vertex colourings. We derive several bounds for infinite graphs, in particular, we prove the general bound D0(G) 6 (G) for an arbitrary infinite graph. Nonetheless, the distinguish- ing index is at most two for many countable graphs, also for the infinite random graph and for uncountable tree-like graphs. We also investigate the concept of the motion of edges and its relationship with the Infinite Motion Lemma.http://www.combinatorics.orgam201

    Group twin coloring of graphs

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    For a given graph GG, the least integer k≥2k\geq 2 such that for every Abelian group G\mathcal{G} of order kk there exists a proper edge labeling f:E(G)→Gf:E(G)\rightarrow \mathcal{G} so that ∑x∈N(u)f(xu)≠∑x∈N(v)f(xv)\sum_{x\in N(u)}f(xu)\neq \sum_{x\in N(v)}f(xv) for each edge uv∈E(G)uv\in E(G) is called the \textit{group twin chromatic index} of GG and denoted by χg′(G)\chi'_g(G). This graph invariant is related to a few well-known problems in the field of neighbor distinguishing graph colorings. We conjecture that χg′(G)≤Δ(G)+3\chi'_g(G)\leq \Delta(G)+3 for all graphs without isolated edges, where Δ(G)\Delta(G) is the maximum degree of GG, and provide an infinite family of connected graph (trees) for which the equality holds. We prove that this conjecture is valid for all trees, and then apply this result as the base case for proving a general upper bound for all graphs GG without isolated edges: χg′(G)≤2(Δ(G)+col(G))−5\chi'_g(G)\leq 2(\Delta(G)+{\rm col}(G))-5, where col(G){\rm col}(G) denotes the coloring number of GG. This improves the best known upper bound known previously only for the case of cyclic groups Zk\mathbb{Z}_k
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