A lower bound and several exact results on the d-lucky number

Abstract

If $\ell :V\left(G\right)\to \mathbb{N}$ is a vertex labeling of a graph $G=\left(V\right(G),E(G\left)\right),$ then the d-lucky sum of a vertex u ∈ V(G) is $d_{\ell }\left(u\right)=d_G\left(u\right)+{\sum }_{v\in N\left(u\right)}\ell \left(v\right)$. The labeling ℓ is a d-lucky labeling if d_ℓ(u) ≠ d_ℓ(v) for every uv ∈ E(G). The d-lucky number η_dl(G) of G is the least positive integer k such that G has a d-lucky labeling V(G) → [k]. A general lower bound on the d-lucky number of a graph in terms of its clique number and related degree invariants is proved. The bound is sharp as demonstrated with an infinite family of corona graphs. The d-lucky number is also determined for the so-called G_m,n-web graphs and graphs obtained by attaching the same number of pendant vertices to the vertices of a generalized cocktail-party graph.

Introduction

In the celebrated paper [14], Karoński et al. asked whether the edges of any graph with no component K₂ can be assigned weights from {1, 2, 3} so that adjacent vertices have different sums of incident edge weights, in other words, such that the resultant vertex weighting is a proper coloring. Although the paper does not use the word “conjecture” for the question, it later (quite naturally) became known as the 1–2–3 Conjecture. The progress on the conjecture until 2012 has been surveyed in [21], while for recent progress see [11], [13] and references therein.

The paper [14] can also be seen as the seed for the investigation of other types of graph labelings in which integers are assigned to some elements of the graph (vertices, edges, or both of them), such that the labeling yields a proper vertex coloring. For us, the most important such labelings are due to Czerwiński et al., who in [7] introduced the concept of the lucky labeling and proposed the conjecture η(G) ≤ χ(G), where η(G) is the lucky number of G (and, of course, χ(G) is the chromatic number of G). For more information on the lucky labelings see [1], [2]. Similar to the lucky number, Chartrand et al. [5] introduced sigma colorings, where the value at a vertex is obtained as the sum of the weights in its neighborhood. Club scheduling problems and hospital planning are real life applications of sigma colorings, cf. [15]. For additional related labelings we refer to [8]. We mention in passing that graph labelings have a variety of applications such as incorporating redundancy in disks, designing drilling machines, creating layouts for circuit boards, and configuring resistor networks, see [23]. Finally, different graph labelings were and are still extensively investigated, we refer to the recent developments on antimagic labelings [4], [16], [17], (d, 1)-labelings [10], [19], and cordial labelings [20], [22], to mention just some of them.

In this paper we are interested in d-lucky labelings that were introduced by Miller et al. [18] as a variant of the lucky labelings as follows. Let $N\left(u\right)=\{v\in V(G):uv\in E(G\left)\right\}$ be the open neighborhood of a vertex u in a graph G. If $\ell :V\left(G\right)\to \mathbb{N}$ is a vertex labeling, then the d-lucky sum of a vertex u ∈ V(G) with respect to ℓ is

$$d_{\ell }\left(u\right)=d_G\left(u\right)+\sum _{v\in N\left(u\right)}\ell \left(v\right),$$

where d_G(u) is the degree of u. The labeling ℓ is a d-lucky labeling if d_ℓ(u) ≠ d_ℓ(v) holds for every pair of adjacent vertices u and v. The d-lucky number η_dl(G) of G is the least positive integer k such that G admits a d-lucky labeling $\ell :V\left(G\right)\to \left[k\right]=\{1,\dots ,k\}$. Lucky labelings are obtained from d-lucky labelings by omitting the additive term d_G(u). A closely related concept of the adjacent vertex distinguishing colorings is defined analogously, except that one adds up the labels in the closed neighborhood of a vertex, see [3], [9].

In the next section we prove a general lower bound on the d-lucky number of a graph in terms of its clique number and related degree invariants. The bound is sharp as demonstrated with an infinite family of corona graphs. The latter result is in turn used in Section 3 to determine the d-lucky number of the so called G_m,n-web graphs. We conclude the paper with the d-lucky number of graphs obtained by attaching the same number of pendant vertices to the vertices of a generalized cocktail-party graph.