1,2 Conjecture—the multiplicative version

doi:10.1016/j.ipl.2008.01.006

Information Processing Letters

Volume 107, Issues 3–4, 31 July 2008, Pages 93-95

https://doi.org/10.1016/j.ipl.2008.01.006 Get rights and content

Abstract

Let us assign positive integers to the edges and vertices of a simple graph G. We consider the colouring of G obtained by assigning to vertex v the product of its weight and those of its adjacent edges. Can we obtain a proper colouring using only weights 1 and 2 for an arbitrary graph G?

We give a positive answer when G is a 3-colourable or complete. We also show that it is enough to use weights 1, 2 and 3 for an arbitrary graph G.

References (5)

L. Addario-Berry et al.
Vertex colouring edge partitions
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L. Addario-Berry et al.
Degree constrained subgraphs

There are more references available in the full text version of this article.

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Several variants of the original problem have been introduced in the literature, such as neighbour-sum-distinguishing total-colouring [6,9] or neighbour-sum-distinguishing arc-colouring [2,3] for instance. Other variants have been obtained by defining the vertex colours by means of product, set or multiset of colours [1,4,5,10]. Consider the strategy for Alice given by the following rules:
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The 1,2-Conjecture for powers of cycles
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A [k]-total-weighting ω of a simple graph G is a mapping $ω : V (G) \cup E (G) \to {1, \dots, k}$ . A [k]-total-weighting ω of G is neighbour-distinguishing if, for each pair of adjacent vertices $u, v \in V (G)$ , the value $ω (u) + \sum_{u w \in E (G)} ω (u w)$ is distinct from $ω (v) + \sum_{v w \in E (G)} ω (v w)$ . The 1,2-Conjecture states that every simple graph G has a neighbour-distinguishing [2]-total-weighting. In this work, we prove that the 1,2-Conjecture is valid for all powers of cycles.
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A proper labeling of a graph is an assignment of integers to some elements of a graph, which may be the vertices, the edges, or both of them, such that we obtain a proper vertex coloring via the labeling subject to some conditions. The problem of proper labeling offers many variants and received a great interest during recent years. We consider the algorithmic complexity of some variants of the proper labeling problems, we present some polynomial time algorithms and NP-completeness results for them.
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In this paper we conjecture that the edges of any non-trivial graph can be weighted with integers 1, 2, 3 in such a way that for every edge uv the product of weights of the edges adjacent to u is different than the product of weights of the edges adjacent to v. It is proven here for cycles, paths, complete graphs and 3-colourable graphs. It is also shown that the edges of every non-trivial graph can be weighted with integers 1, 2, 3, 4 in such a way that the adjacent vertices have different products of incident edge weights.
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Suppose the vertices of a graph G were labeled arbitrarily by positive integers, and let $S (v)$ denote the sum of labels over all neighbors of vertex v. A labeling is lucky if the function S is a proper coloring of G, that is, if we have $S (u) \neq S (v)$ whenever u and v are adjacent. The least integer k for which a graph G has a lucky labeling from the set ${1, 2, \dots, k}$ is the lucky number of G, denoted by $η (G)$ .
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Note1,2 Conjecture—the multiplicative version

Abstract

J. Combin. Theory Ser. B

Note
$1, 2$ Conjecture—the multiplicative version