Distant total irregularity strength of graphs via random vertex ordering

TL;DR

This paper establishes an improved probabilistic bound on the total irregularity strength of graphs for vertices within a certain distance, advancing understanding of graph colorings related to the 1-2-Conjecture.

Contribution

It combines probabilistic and combinatorial methods to prove a tighter upper bound on the total irregularity strength for any graph and distance parameter.

Findings

Proves that ${ m ts}_r(G) o (2+o(1))\Delta^{r-1}$ for all graphs and r ≥ 2.

Improves previous bound of ${ m ts}_r(G) o 3\Delta^{r-1}$.

Uses a novel combination of probabilistic and combinatorial techniques.

Abstract

Let $c:V\cup E\to\{1,2,\ldots,k\}$ be a (not necessarily proper) total colouring of a graph $G=(V,E)$ with maximum degree $\Delta$. Two vertices $u,v\in V$ are sum distinguished if they differ with respect to sums of their incident colours, i.e. $c(u)+\sum_{e\ni u}c(e)\neq c(v)+\sum_{e\ni v}c(e)$. The least integer $k$ admitting such colouring $c$ under which every $u,v\in V$ at distance $1\leq d(u,v)\leq r$ in $G$ are sum distinguished is denoted by ${\rm ts}_r(G)$. Such graph invariants link the concept of the total vertex irregularity strength of graphs with so called 1-2-Conjecture, whose concern is the case of $r=1$. Within this paper we combine probabilistic approach with purely combinatorial one in order to prove that ${\rm ts}_r(G)\leq (2+o(1))\Delta^{r-1}$ for every integer $r\geq 2$ and each graph $G$, thus improving the previously best result: ${\rm ts}_r(G)\leq 3\Delta^{r-1}$.

Equations

$$w_c(v):=c(v)+\sum _{u\in N(v)}c(uv)$$

$${\mathrm{ts}}_r(G)\le 2{\Delta }^{r-1}+3{\Delta }^{r-\frac{4}{3}}{\ln }^2\Delta +4,$$

$${\mathrm{ts}}_r(G)\le (2+o(1)){\Delta }^{r-1}$$

$$ep(D+1)\le 1,$$

$$\text{Pr}(\mathrm{BIN}(n,p)>np+t)<e^{-\frac{t^2}{3np}}and\text{Pr}(\mathrm{BIN}(n,p)<np-t)<e^{-\frac{t^2}{2np}}\le e^{-\frac{t^2}{3np}}$$

$$S$$

$$B$$

$$I$$

$$R$$

$$\mathbf{Pr}(A_{v,1})$$

$$\mathbf{Pr}(b_{-}(v)<X_{v}b(v)-\sqrt{X_{v}b(v)}\ln \Delta |X_v=x)$$

$$\mathbf{Pr}(A_{v,2})\le \mathbf{Pr}(b_{-}(v)<X_{v}b(v)-\sqrt{X_{v}b(v)}\ln \Delta )\le \underset{0}{\overset{1}{\int }}\frac{1}{{\Delta }^{3r}}dx=\frac{1}{{\Delta }^{3r}}.$$

$$\mathbf{Pr}(d_{-}^r(v)>X_{v}D(v){\Delta }^{r-2}+\sqrt{X_{v}D(v){\Delta }^{r-2}}\ln \Delta \wedge v\in R|X_v=x)=0.$$

$$\mathbf{Pr}(d_{-}^r(v)>X_{v}D(v){\Delta }^{r-2}+\sqrt{X_{v}D(v){\Delta }^{r-2}}\ln \Delta \wedge v\in R|X_v=x)$$

$$\mathbf{Pr}(A_{v,3})\le \underset{0}{\overset{1}{\int }}\frac{1}{{\Delta }^{3r}}dx=\frac{1}{{\Delta }^{3r}}.$$

$$K={\Delta }^{r-1}+\lceil {\Delta }^{r-\frac{4}{3}}{\ln }^2\Delta \rceil \mathrm{and}k=\lceil {\Delta }^{r-\frac{4}{3}}{\ln }^2\Delta \rceil ,$$

$$0\le w_f(v)-w_{c_t}(v)\le K.$$

$$b_{-}(v)K+(d-b_{-}(v))k$$

$$|N_{-}^r(v)|$$

$${\mathrm{ts}}_r(G)\le (1+o(1)){\Delta }^{r-1}.$$

Full text

1 Introduction

The cornerstone of the field of vertex distinguishing graph colourings is the graph invariant called irregularity strength. For a graph $G=(V,E)$ it is usually denoted by $s(G)$ and can be defined as the least integer $k$ so that we may construct an irregular multigraph, i.e. a multigraph with pairwise distinct degrees of all vertices, of $G$ by multiplying its edges, each at most $k$ times (including the original one), see [8]. This study thus originated from the basic fact that no graph $G$ with more than one vertex is irregular itself and related research on possible alternative definitions of an irregular graph, see e.g. [7]. Equivalently, $s(G)$ is also defined as the least $k$ so that there exists an edge colouring $c:E\to\{1,2,\ldots,k\}$ such that for every pair $u,v\in E$, $u\neq v$, the sum of colours incident with $u$ is distinct from the sum of colours incident with $v$. Note that $s(G)$ exists only for graphs without isolated edges and with at most one isolated vertex. It is known that $s(G)\leq n-1$, where $n=|V|$, for all such graphs, except for $K_{3}$, see [1, 21]. This tight upper bound can however be improved in the case of graphs with minimum degree $\delta\geq 1$ to $s(G)\leq 6\lceil\frac{n}{\delta}\rceil$ (what yields a better result whenever $\delta>12$ and for $\delta\in[7,12]$ if $n$ is larger than a small constant dependent on $\delta$), see [16], and to $s(G)\leq(4+o(1))\frac{n}{\delta}+4$ for graphs with $\delta\geq n^{0.5}\ln n$, see [19]. Many interesting results, concepts and open problems concerning this graph invariant can also be found e.g. in [6, 9, 10, 11, 12, 13, 16, 18, 25], and many others.

In [5], Bača et al. introduced a total version of the concept above. Given any graph $G=(V,E)$ and a (not necessarily proper) total colouring $c:V\cup E\to\{1,2,\ldots,k\}$, let

[TABLE]

denote the weight of any vertex $v\in V$, which shall also be called the sum at $v$ and denoted simply by $w(v)$ in cases when $c$ is unambiguous from context. The least $k$ for which there exists such colouring with $w(u)\neq w(v)$ for every $u,v\in V$, $u\neq v$, is called the total vertex irregularity strength of $G$ and denoted by ${\rm tvs}(G)$. In [5], among others, it was proved that for every graph $G$ with $n$ vertices, $\lceil\frac{n+\delta}{\Delta+1}\rceil\leq{\rm tvs}(G)\leq n+\Delta-2\delta+1$. Up to know the best upper bounds (for graphs with $\delta>3$) assert that ${\rm tvs}(G)\leq 3\lceil\frac{n}{\delta}\rceil+1$, see [3], and ${\rm tvs}(G)\leq(2+o(1))\frac{n}{\delta}+4$ for $\delta\geq n^{0.5}\ln n$, see [20]. Many other results e.g. for particular graph families can also be found in [4, 22, 26, 29] and other papers.

In this article we consider a distant generalization of ${\rm tvs}(G)$ from [23], motivated among others by the study on distant chromatic numbers, see e.g. [17] for a survey concerning these. For any positive integer $r$, two distinct vertices at distance at most $r$ in $G$ shall be called $r$-neighbours. We denote by $N^{r}(v)$ the set of all $r$-neigbours of any $v\in V$ in $G$, and set $d^{r}(v)=|N^{r}(v)|$. The least integer $k$ for which there exists a total colouring $c:V\cup E\to\{1,2,\ldots,k\}$ such that there are no $r$-neighbours $u,v$ in $G$ which are in conflict, i.e. with $w(u)=w(v)$ (cf. (1)), we call the $r$-distant total irregularity strength of $G$, and denote by ${\rm ts}_{r}(G)$. It is known that ${\rm ts}_{r}(G)\leq 3\Delta^{r-1}$ for every graph $G$ , see [23], also for a comment implying that a general upper bound for ${\rm ts}_{r}(G)$ cannot be (much) smaller than $\Delta^{r-1}$. In this paper we combine the probabilistic method with algorithmic approach similar to those in e.g. [3, 15, 20, 23] to prove that in fact ${\rm ts}_{r}(G)\leq(2+o(1))\Delta^{r-1}$ (for $r\geq 2$).

Theorem 1

For every integer $r\geq 2$ there exists a constant $\Delta_{0}$ such that for each graph $G$ with maximum degree $\Delta\geq\Delta_{0}$,

[TABLE]

hence

[TABLE]

for all graphs.

It is also worth mentioning that the case of $r=1$ was introduced and considered separately in [27], where the well known 1-2-Conjecture concerning this invariant was introduced. It is known that ${\rm ts}_{1}(G)\leq 3$ for all graphs, see Theorem 2.8 in [15], even in case of a natural list generalization of the problem, see [30], though it is believed that the upper bound of $2$ should make the optimal general upper bound in both cases, see [27, 28, 31].

We also refer a reader to [24] to see an improvement of a similar probabilistic flavor for the upper bound from [23] on the correspondent of ${\rm ts}_{r}(G)$ concerning the case of edge colourings exclusively.

2 Probabilistic Tools

We shall use probabilistic approach in the first part of the proof of Theorem 1, basing on the Lovász Local Lemma, see e.g. [2], combined with the Chernoff Bound, see e.g. [14] (Th. 2.1, page 26). We recall these below.

Theorem 2** (The Local Lemma)**

Let $A_{1},A_{2},\ldots,A_{n}$ be events in an arbitrary probability space. Suppose that each event $A_{i}$ is mutually independent of a set of all the other events $A_{j}$ but at most $D$, and that ${\rm\emph{{Pr}}}(A_{i})\leq p$ for all $1\leq i\leq n$. If

[TABLE]

then ${\rm\emph{{Pr}}}\left(\bigcap_{i=1}^{n}\overline{A_{i}}\right)>0$.

Theorem 3** (Chernoff Bound)**

For any $0\leq t\leq np$,

[TABLE]

where ${\rm BIN}(n,p)$ is the sum of $n$ independent Bernoulli variables, each equal to $1$ with probability $p$ and [math] otherwise.

Note that if $X$ is a random variable with binomial distribution ${\rm BIN}(n,p)$ where $n\leq k$, then we may still apply the Chernoff Bound above, even if we do not know the exact value of $n$, to prove that $\mathbf{Pr}(X>kp+t)<e^{-\frac{t^{2}}{3kp}}$ (for $t\leq\lfloor k\rfloor p$).

3 Proof of Theorem 1

Fix an integer $r\geq 2$. Within our proof we shall not specify $\Delta_{0}$. Instead, we shall assume that $G=(V,E)$ is a graph with sufficiently large maximum degree $\Delta$, i.e. large enough so that all inequalities below are fulfilled.

We first partition $V$ into a subset of vertices with relatively small degrees and a subset of those with big degrees:

[TABLE]

Moreover, for every $v\in V$, we denote: $S(v)=N(v)\cap S$, $s(v)=|S(v)|$, $B(v)=N(v)\cap B$, $b(v)=|B(v)|$.

Now we randomly order the vertices of $V$ into a sequence. For this goal, associate with every vertex $v\in V$ a random variable $X_{v}\sim U[0,1]$ having the uniform distribution on $[0,1]$ where all these random variables $X_{v}$, $v\in V$ are independent, or in other words pick a (real) number uniformly at random from the interval $[0,1]$ and associate it with $v$ for every $v\in V$. Note that with probability one all these numbers are pairwise distinct. In such a case, these independent random variables uniquely define a natural ordering $v_{1},v_{2},\ldots,v_{n}$ of the vertices in $V$ where $X_{v_{i}}<X_{v_{j}}$ if and only if $1\leq i<j\leq n$.

For every vertex $v\in V$, any its neighbour or $r$-neighbour $u$ which precedes $v$ in the obtained ordering of the elements of $V$ shall be called a backward neighbour or $r$-neighbour, resp., of $v$. Analogously, the remaining ones shall be called forward neighbours or $r$-neighbours, resp., of $v$, while the edges joining $v$ with its forward or backward neighbours shall be referred to as forward or backward edges, resp., as well. Also, for any subset $W\subset V$, let $N_{-}(v)$, $N_{-}^{r}(v)$, $N_{W}^{r}(v)$ denote the sets of all backward neighbours, backward $r$-neighbours and $r$-neighbours in $W$ of $v$, respectively. Set $d_{-}^{r}(v)=|N_{-}^{r}(v)|$, $d_{W}^{r}(v)=|N_{W}^{r}(v)|$, and let $b_{-}(v)$ denote the number of backward neighbours of $v$ which belong to $B(v)$.

Denote $D(v)=\sum_{u\in N(v)}d(u)$ and note that $d^{r}(v)\leq D(v)\Delta^{r-2}\leq d(v)\Delta^{r-1}$.

Let us also partition $V$ into a subset $I$ consisting of initial vertices of the obtained sequence and the remaining part $R$:

[TABLE]

Lemma 4

With positive probability, the obtained ordering has the following features for every vertex $v$ in $G$ with $b(v)\geq\Delta^{\frac{1}{3}}\ln\Delta$:

• $F_{1}$:

$d^{r}_{I}(v)\leq 2d(v)\Delta^{r-\frac{4}{3}}\ln\Delta$;

• $F_{2}$:

if $v\in R$, then: $b_{-}(v)\geq X_{v}b(v)-\sqrt{X_{v}b(v)}\ln\Delta$;

• $F_{3}$:

if $v\in R$, then: $d^{r}_{-}(v)\leq X_{v}D(v)\Delta^{r-2}+\sqrt{X_{v}D(v)\Delta^{r-2}}\ln\Delta$.

Proof 1

For every vertex $v\in V$ of degree $d$ in $G$ and with $b(v)\geq\Delta^{\frac{1}{3}}\ln\Delta$ (hence also $d\geq\Delta^{\frac{1}{3}}\ln\Delta$), let $A_{v,1}$ denote the event that $d^{r}_{I}(v)>2d\Delta^{r-\frac{4}{3}}\ln\Delta$, let $A_{v,2}$ be the event that $v$ belongs to $R$ and $b_{-}(v)<X_{v}b(v)-\sqrt{X_{v}b(v)}\ln\Delta$, and let $A_{v,3}$ denote the event that $v$ belongs to $R$ and $d^{r}_{-}(v)>X_{v}D(v)\Delta^{r-2}+\sqrt{X_{v}D(v)\Delta^{r-2}}\ln\Delta$.

As $|N^{r}(v)|\leq d\Delta^{r-1}$ and for each $u\in N^{r}(v)$, the probability that $u$ belongs to $I$ equals $\frac{\ln\Delta}{\Delta^{\frac{1}{3}}}$, then by the Chernoff Bound (and the comment below it),

[TABLE]

Subsequently note that for any $x\in[0,1]$:

[TABLE]

where the last inequality follows by the Chernoff Bound if $\sqrt{xb(v)}\ln\Delta\leq xb(v)$, while it is trivial otherwise. Hence,

[TABLE]

For the sake of analyzing $A_{v,3}$, note now first that for $x\in[0,\frac{\ln\Delta}{\Delta^{\frac{1}{3}}})$,

[TABLE]

On the other hand, analogously as above, for $x\in[\frac{\ln\Delta}{\Delta^{\frac{1}{3}}},1]$:

[TABLE]

where the last inequality follows by the Chernoff Bound, as $x\geq\frac{\ln\Delta}{\Delta^{\frac{1}{3}}}$ and $b(v)\geq\Delta^{\frac{1}{3}}\ln\Delta$ (where $D(v)\geq b(v)\Delta^{\frac{2}{3}}$) imply that $\sqrt{xD(v)\Delta^{r-2}}\ln\Delta\leq xD(v)\Delta^{r-2}$. Hence, by (5) and (6),

[TABLE]

Note that each event $A_{v,i}$ is mutually independent of all other events except those $A_{u,j}$ with $u$ at distance at most $2r$ from $v$, $i,j\in\{1,2,3\}$, i.e., at most $3\Delta^{2r}+2$ events. Thus, as by (3), (4) and (7), the probability of each such event is bounded from above by $\Delta^{-3r}$, by the Lovász Local Lemma, with positive probability none of the events $A_{v,i}$ with $v\in V$ (and $b(v)\geq\Delta^{\frac{1}{3}}\ln\Delta$) and $i\in\{1,2,3\}$ appears. \qed

Let $v_{1},v_{2},\ldots,v_{n}$ be the ordering of the vertices of $V$ guaranteed by Lemma 4. Set

[TABLE]

and assign initial colour $1$ to all the vertices and initial colour $K+1$ to all the edges of $G$. We shall construct our final colouring $f:V\cup E\to\{1,2,\ldots,2K+k+1\}$ using an algorithm within which we shall be analyzing the consecutive vertices in the ordering (starting from $v_{1}$). Denote by $c_{t}(a)$ the contemporary colour of every $a\in V\cup E$ at every stage of the ongoing algorithm (hence initially $c_{t}(v)=1$ and $c_{t}(e)=K+1$ for every $v\in V$ and $e\in E$). The final target sum of every vertex $v\in V$, $w_{f}(v)$, shall be chosen the moment $v$ is analyzed. For every $v\in V$, ever since $w_{f}(v)$ is chosen, we shall require so that

[TABLE]

We shall admit at most two alterations of the colour for every edge in $E$ - only when any of its ends is being analyzed (vertex colours shall be adjusted at the end of the algorithm). For every currently analyzed vertex $v$ and its neighbour $u\in N(v)$, we admit the following alterations of the colour of $e=uv$ (the moment $v$ is analyzed):

adding $0,1,\ldots,K-1$ or $K$ if $e$ is a forward edge of $v$, $v\in S$ and $u\in B$,

• 2.

adding $0,1,\ldots,k-1$ or $k$ if $e$ is a forward edge of $v$ (and $v\in B$ or $u\in S$),

• 3.

adding $-K,-K+1,\ldots,K-1$ or $K$ if $e$ is a backward edge of $v$ and $u\in B$,

• 4.

adding $-k,-k+1,\ldots,k-1$ or $k$ if $e$ is a backward edge of $v$ and $u\in S$,

so that afterwards (8) is fulfilled for every vertex $u\in N_{-}(v)$ and for $v$ (after processing all edges incident with $v$). Note that the admitted alterations guarantee that $c_{t}(e)\in\{1,2,\ldots,2K+k+1\}$ for every $e\in E$ at every stage of the construction.

Suppose we are about to analyze a vertex $v=v_{i}$, $i\in\{1,2,\ldots,n\}$, and thus far all our requirements have been fulfilled. We shall show that in every case the admitted alterations on the edges incident with $v$ provide us more options for $w_{c_{t}}(v)$ than there are backward $r$-neighbours of $v$, and hence one of this options can be fixed as $w_{f}(v)$ so that this value is distinct from every $w_{f}(u)$ already fixed for any $u\in N^{r}_{-}(v)$. Denote the degree of $v$ by $d$, and assume that $d>0$ (otherwise, we set $w_{f}(v)=1$):

If $v\in I$, $v\in B$ and $b(v)\geq\Delta^{\frac{1}{3}}\ln\Delta$, then the admitted alterations provide at least $dk\geq d\Delta^{r-\frac{4}{3}}\ln^{2}\Delta$ available options for $w_{c_{t}}(v)$. As by $F_{1}$ (from Lemma 4), $|N^{r}_{-}(v)|\leq d_{I}^{r}(v)\leq 2d\Delta^{r-\frac{4}{3}}\ln\Delta<d\Delta^{r-\frac{4}{3}}\ln^{2}\Delta$, at least one of these available options is distinct from all $w_{f}(u)$ with $u\in N^{r}_{-}(v)$.

• 2.

If $v\in S$, then the admitted alterations provide at least $s(v)k+b(v)K\geq s(v)\Delta^{r-\frac{4}{3}}\ln^{2}\Delta+b(v)(\Delta^{r-1}+\Delta^{r-\frac{4}{3}}\ln^{2}\Delta)$ available options for $w_{c_{t}}(v)$. On the other hand, $|N^{r}_{-}(v)|\leq d^{r}(v)\leq D(v)\Delta^{r-2}\leq(s(v)\Delta^{\frac{2}{3}}+b(v)\Delta)\Delta^{r-2}$, hence at least one of these available options is distinct from all $w_{f}(u)$ with $u\in N^{r}_{-}(v)$.

• 3.

If $v\in B$ and $b(v)<\Delta^{\frac{1}{3}}\ln\Delta$, then the admitted alterations provide at least $dk\geq d\Delta^{r-\frac{4}{3}}\ln^{2}\Delta$ available options for $w_{c_{t}}(v)$. On the other hand, analogously as in the case above, $|N^{r}_{-}(v)|\leq d^{r}(v)\leq s(v)\Delta^{\frac{2}{3}}\Delta^{r-2}+b(v)\Delta^{r-1}<d\Delta^{r-\frac{4}{3}}+\Delta^{r-\frac{2}{3}}\ln\Delta<d\Delta^{r-\frac{4}{3}}\ln^{2}\Delta$, as $v\in B$ implies that $d\geq\Delta^{\frac{2}{3}}$. We thus have at least one option available for $v$ distinct from all $w_{f}(u)$ with $u\in N^{r}_{-}(v)$.

• 4.

If $v\in R$, $v\in B$ and $b(v)\geq\Delta^{\frac{1}{3}}\ln\Delta$, then by $F_{2}$ the number of available options for $w_{c_{t}}(v)$ via admitted alterations of colours of the edges incident with $v$ is not smaller than:

[TABLE]

(where the last inequality follows by the fact that $d\geq\Delta^{\frac{2}{3}}$). This number is however greater than the number of backward $r$-neighbours of $v$, as by $F_{3}$,

[TABLE]

Thus in all cases there is at least one available sum, say $w^{*}$, for $v$ which is distinct from all $w_{f}(u)$ with $u\in N^{r}_{-}(v)$. We then set $w_{f}(v)=w^{*}$ and perform the admitted alterations on the edges incident with $v$ so that $w_{c_{t}}(v)=w^{*}$ afterwards.

By our construction, after analyzing $v_{n}$, all $w_{f}(v_{i})$ are fixed for $i=1,\ldots,n$ so that $w_{f}(u)\neq w_{f}(v)$ whenever $u$ and $v$ are $r$-neighbours in $G$ and (8) holds for every $v\in V$. We then modify (if necessary) the colour of every vertex $v$ by adding to it the integer $w_{f}(v)-w_{c_{t}}(v)$, completing the construction of the desired total colouring $f$ of $G$ (by setting $f(a)=c_{t}(a)$ for every $a\in V\cup E$ afterwards). Note that $1\leq f(e)\leq 2K+k+1$ for every $e\in E$ and, by (8), $1\leq f(v)\leq K+1$ for every $v\in V$, hence the thesis follows. \qed

4 Remarks

We have put an effort to optimize the second order term from the upper bound in (2), up to a constant and a power in the logarithmic factor, which could still be slightly improved (at the cost of the clarity of presentation). Nevertheless, some multiplicative poly-logarithmic (in $\Delta$) factor seems unavoidable in this term within our approach.

We conclude by posing a conjecture, which to our believes expresses a true asymptotically optimal upper bound for the investigated parameters.

Conjecture 5

For every integer $r\geq 2$ and each graph $G$ with maximum degree $\Delta$,

[TABLE]