We consider a proper coloring c of edges and vertices in a simple graph and the sum f (v) of colors of all the edges incident to v and the color of a vertex v. We say that a coloring c distinguishes adjacent vertices by sums, if every two adjacent vertices have different values of f . We conjecture that + 3 colors suffice to distinguish adjacent vertices in any simple graph. In this paper we show that this holds for complete graphs, cycles, bipartite graphs, cubic graphs and graphs with maximum degree at most three.
The distinguishing index of a graph G, denoted by D (G), is the least number of colours in an edge colouring of G not preserved by any non-trivial automorphism. We characterize all connected graphs G with D (G) ≥ ∆(G). We show that D (G) ≤ 2 if G is a traceable graph of order at least seven, and D (G) ≤ 3 if G is either claw-free or 3-connected and planar. We also investigate the Nordhaus-Gaddum type relation: 2 ≤ D (G) + D (G) ≤ max{∆(G), ∆(G)} + 2 and we confirm it for some classes of graphs.
The distinguishing index $D^\prime(G)$ 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 $D^\prime(G)\leq\Delta(G)$ for an arbitrary infinite graph. Nonetheless, the distinguishing 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.
The distinguishing index D (G) of a graph G is the least natural number d such that G has an edge colouring with d colours that is only preserved by the identity automorphism. In this paper we investigate the distinguishing index of the Cartesian product of connected finite graphs. We prove that for every k ≥ 2, the k-th Cartesian power of a connected graph G has distinguishing index equal 2, with the only exception D (K 2 2) = 3. We also prove that if G and H are connected graphs that satisfy the relation 2 ≤ |G| ≤ |H| ≤ 2 |G| 2 G − 1 − |G| + 2, then D (G2H) ≤ 2 unless G2H = K 2 2 .
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