# On colour-blind distinguishing colour pallets in regular graphs

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## Abstract

Consider a graph \(G=(V,E)\) and a colouring of its edges with \(k\) colours. Then every vertex \(v\in V\) is associated with a ‘pallet’ of incident colours together with their frequencies, which sum up to the degree of \(v\). We say that two vertices have distinct pallets if they differ in frequency of at least one colour. This is always the case if these vertices have distinct degrees. We consider an apparently the worse case, when \(G\) is regular. Suppose further that this coloured graph is being examined by a person who cannot name any given colour, but distinguishes one from another. Could we colour the edges of \(G\) so that a person suffering from such colour-blindness is certain that colour pallets of every two adjacent vertices are distinct? Using the Lopsided Lovász Local Lemma, we prove that it is possible using 15 colours for every \(d\)-regular graph with \(d\ge 960\).

### Keywords

Neighbour-distinguishing colouring Lopsided Lovász Local Lemma Colour pallet## 1 Distinguishing colour pallets by colour-blind

Consider a simple graph \(G=(V,E)\) and an edge colouring \(c:E\rightarrow \{1,2,\ldots ,k\}\), not necessarily proper. Such colouring is called *neighbour distinguishing* (or *vertex colouring*, see e.g., Addario-Berry et al. 2005) if for every edge \(uv\in E\), the multiset of colours incident with \(u\) is distinct from the multiset of colours incident with \(v\). In other words, if for every vertex \(v\) we set \(\overline{c}(v)=(a_1,\ldots ,a_k)\), where \( a_i =|\{w:wv\in E, c (wv)=i\}|\) for \(i=1,\ldots ,k\), then the colouring \(c\) is neighbour distinguishing if \(\overline{c}(u)\ne \overline{c}(v)\) for each edge \(uv\) of \(G\). Clearly, one can find such colouring if a graph contains no isolated edges, e.g., by painting each edge differently. Karoński et al. (2004) first proved that in fact a finite number of 183 colours are always sufficient, or even 30 if the minimum degree \(\delta \) of \(G\) is at least \(10^{99}\). This was then greatly improved by Addario-Berry et al. (2005), who showed that four colours are sufficient, and these can be decreased to three if \(\delta \ge 1,000\).

Suppose now that such coloured graph is examined by a colour-blind person, i.e., somebody who cannot name colours but distinguishes one from another. Can such individual ‘distinguish neighbours’ then? The answer is affirmative in many cases. It is due to the fact that given a set of coloured edges, they are able to divide it into monochromatic subsets and count their cardinalities. Given any sequence \(\overline{c}(v)=(a_1,\ldots ,a_k)\), let us re-order it non-decreasingly. The obtained sequence \(c^*(v)=(d_1,\ldots ,d_k)\) we shall call a *pallet* of \(v\). Note that there is a bijection between the set of all possible pallets one may obtain for a vertex \(v\) of degree \(d\) and the set of all \(k\)-partitions of the integer \(d\), i.e., the set \(P(d,k)=\{(d_1,d_2,\ldots ,d_k)\in \mathbb N ^k:d_1+\cdots +d_k=d\;\mathrm{and}\; 0\le d_i\le d_{i+1}\; \mathrm{for}\;i =1,\ldots , k-1 \}\). We say that a *colour-blind person can distinguish neighbours* in our colouring \(c:E\rightarrow \{1,\ldots ,k\}\) if \(c^*(u)\ne c^*(v)\) for every edge \(uv\in E\). The smallest integer \(k\) for which such colouring exists is called the *colour-blind index* of \(G\), and is denoted by \(\mathrm{dal}(G)\). This notion refers to the English chemist John Dalton, who in \(1798\) wrote the first paper on colour-blindness. In fact, because of Dalton’s work, the condition is often called *daltonism*.

It has to be noted that this parameter is undefined for some classes of graphs, in particular we must exclude graphs with isolated edges. However, thus far all known graphs with undefined colour-blind index have minimum degree at most three, see Kalinowski et al. for details. It has been proved there that given a fixed \(R> 1\), there always exists \(\delta _R\) such that \(\mathrm{dal}(G)\le 6\) for every graph with maximum degree \(\Delta \le R\delta \), provided that \(\delta \ge \delta _R\). Unfortunately \(\delta _R\) tends to infinity along with \(R\). It is thus not even known whether graphs with \(\delta \ge \delta _0\) have well defined colour-blind index for any constant \(\delta _0\), though Kalinowski et al. conjecture that it is so (maybe even with \(\delta _0=4\)). Situation with this mysterious parameter changes if we restrict ourselves to regular graphs exclusively. Using a Lovász Local Lemma, Kalinowski et al. proved that \(\mathrm{dal}(G)\le 6\) for every \(d\)-regular graph \(G\) if its degree is greater than a huge constant, namely, if \(d\ge 2\times 10^7\). An application of the probabilistic method in this context meets unusual obstacles. Unlike in many other similar problems, increasing the number of colours, ‘helps’ only until a certain point. Then the probability of a ‘bad event’ that vertices are indistinguishable for a colour-blind person (e.g., when the edge colouring is proper) grows. In this paper we optimize this probabilistic approach in order to significantly reduce the threshold for \(d\) at the cost of a few more colours. We shall thus prove the following theorem.

**Theorem 1**

The proof is based on the following variation of the Lovász Local Lemma, due to Erdős and Spencer (1991), sometimes referred to as the ‘Lopsided’ Local Lemma. We recall its symmetric versions from Alon and Spencer (2000) (see Corollary 5.1.2 and the comments below).

**Theorem 2**

## 2 Proof of Theorem 1

### 2.1 Random process and dependency digraph

Suppose we are given a \(d\)-regular graph \(G=(V,E)\) with \(d\ge 960\). For each edge \(e\,{\in }\, E\) we independently and randomly choose a colour from the available set \(\{1,2,\ldots ,15\}\), each with equal probability, and denote it by \(c(e)\). In other words, the edges of \(G\) are associated with a set of independent random variables \((X_e)_{e\in E}\), each taking one of the values \(1,2,\ldots ,15\) with probability \(1/15\). Outcomes for these determine an edge colouring of \(G\), each occurring with probability \(1/15^{|E|}\) within the associated product probability space. By a *bad event*\(A_{e}\) in our random process of generating \(c\) we shall mean obtaining \(c^*(u)= c^*(v)\) for some edge \(e=uv\in E\). If no bad event occurs, the corresponding colouring shall meet our requirements. We thus need to show that the probability of the event \(\bigcap _{e\in E}\overline{A_e}\) is positive in our probability space.

*dependency digraph*, in the following manner. Let \(\mathcal A =\{A_{e}: e\in E\}\). Now for every edge \(e=uv\) (i.e. \(e=\{u,v\}\)) of \(G\), we arbitrarily choose one of its end vertices, say \(v\). Equivalently, we choose an orientation \(\overrightarrow{e}=(u,v)\) of every edge \(e\in E\), and the obtained orientation of \(G\) we denote by \(\overrightarrow{G}=(V,\overrightarrow{E})\). Then for every edge \(e\in E\) with orientation \(\overrightarrow{e}=(u,v)\), we draw an arc between \(A_e\) and every event \(A_{e^{\prime }}\) such that \(e^{\prime }\) is at distance at most \(2\) from \(v\) in a graph \(G-e\) (where an edge incident with a vertex is at distance \(1\) from it), i.e., \(e^{\prime }\) is incident with some neighbour of \(v\) different from \(u\). The set of all such arcs we denote by \(\mathcal E \). Note that then

### 2.2 Conditional probability of a bad event

*repetition*if \(d_i=d_j\) for some \(j < i\) (hence \(r(d_1,\ldots ,d_{15})=15-q\)), and note that by (6),

*at most*\(r\) repetitions, \(r=0,\ldots ,14\).

### 2.3 Partitions without repetitions

Given a not necessarily monotone sequence of non-negative integers \(k_1,\ldots ,k_{15}\) summing up to \(d\), by *tightening* its two elements \(k_i,k_j\) satisfying \(k_j\ge k_i+2\) we shall mean substituting these with the elements \(k_i+1\) and \(k_j-1\). Note that such operation always ‘increase the value’ of \(\bigg (\begin{array}{c}d\\ k_1\ldots k_{15}\end{array}\bigg )\), since if without lost of generality, \(i=1\) and \(j=15\), i.e, \(k_{15}\ge k_1+2>k_1+1\), then \(\bigg (\begin{array}{c}d\\ k_1+1 k_2\ldots k_{14} k_{15}-1\end{array}\bigg ) = \frac{k_{15}}{k_1+1} \bigg (\begin{array}{c}d\\ {k_1\ldots k_{15}}\end{array}\bigg )>{d\atopwithdelims (){k_1\ldots k_{15}}}\). Moreover, the minimum and maximum of this sequence shall be called its *left* and *right borders*, resp., and every integer which does not appear in the sequence, but is between its borders shall be called a *gap*.

*majorization inequality*, applied for the function \(f(x)=\log _{0.5}x\).

**Theorem 3**

### 2.4 Partitions with one repetition

### 2.5 Partitions with at least two repetitions

Assume now that \(i\ge 2\). Since \(d_1\) and \(d_1+14-i\) are the borders of \(c^*_{d,i}\), then analogously as above, \(d\le d_1+(d_1+1)+\cdots +(d_1+14-i)+i(d_1+14-i) = 15d_1+\frac{(15-i)(14-i)}{2}+i(14-i) = 15d_1+\frac{(15+i)(14-i)}{2} \le 15d_1+102\) (for \(i\ge 2\)), hence \(d_1 \ge \frac{d-102}{15}\ge \frac{858}{15}\).

## Acknowledgments

This research was partly supported by the National Science Centre Grant No. DEC-2011/01/D/ST1/04154 and by the Polish Ministry of Science and Higher Education. I enclose special thanks to Andrzej Żak for fruitful discussions in 306A.

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