Total Distinguishing Chromatic Number

• Total distinguishing chromatic number is the minimum number of colors in a proper total coloring that ensures adjacent vertices have different incident color-sets.
• Key methods include deterministic recoloring for low-degree vertices and probabilistic techniques for high-degree vertices to achieve a Δ(G)+O(1) bound.
• The parameter bridges total coloring and vertex distinguishing techniques, with a conjectured optimum of Δ(G)+3 and implications for sharper total-coloring bounds.

The total distinguishing chromatic number, in the sense treated in "The adjacent vertex distinguishing total chromatic number," is the minimum number of colors in a proper total coloring of a graph $G$ such that every pair of adjacent vertices receives different incident color-sets (Coker et al., 2010). For a proper total coloring $\varphi$, the color-set of a vertex $v$ is

$$C_{\varphi}(v)=\{\varphi(v)\}\cup\{\varphi(vw):w\in N(v)\},$$

and the requirement is that $C_{\varphi}(u)\neq C_{\varphi}(v)$ whenever $uv\in E(G)$ (Coker et al., 2010). In that paper the parameter is called the adjacent vertex distinguishing total chromatic number and is denoted $\chi_{at}(G)$; in this usage, it is exactly the parameter meant by “total distinguishing chromatic number” (Coker et al., 2010).

1. Formal definition and basic inequalities

Let $G=(V,E)$ be a finite simple graph. A map

$\varphi:V\cup E\to [k]$

is a proper total $k$-coloring if adjacent vertices receive different colors, adjacent edges receive different colors, and each vertex receives a color different from every incident edge (Coker et al., 2010). Equivalently, $\varphi$0 is a proper vertex coloring, $\varphi$1 is a proper edge coloring, and no vertex shares a color with an incident edge (Coker et al., 2010).

Given such a coloring, the color-set

$\varphi$2

records the color on $\varphi$3 together with the colors on all edges incident with $\varphi$4 (Coker et al., 2010). A coloring is adjacent vertex distinguishing if $\varphi$5 for every edge $\varphi$6, and the least $\varphi$7 for which such a coloring exists is

$\varphi$8

the adjacent vertex distinguishing total chromatic number (Coker et al., 2010).

This parameter sits between ordinary total coloring and several trivial upper bounds. Since every adjacent vertex distinguishing total coloring is in particular a proper total coloring,

$\varphi$9

where $v$0 is the total chromatic number (Coker et al., 2010). Also, any graph satisfies

$v$1

because a vertex of maximum degree must avoid the colors of all its incident edges (Coker et al., 2010). On the other hand, if one uses disjoint color sets for a proper vertex coloring and a proper edge coloring, then adjacent vertices are automatically distinguished, yielding

$v$2

Using Brooks’ theorem and Vizing’s theorem, if $v$3 is not a complete graph or an odd cycle, this gives

$v$4

as a straightforward bound (Coker et al., 2010).

2. Global bounds and the $v$5 conjecture

The central theorem of (Coker et al., 2010) is that $v$6 differs from $v$7 by at most an absolute constant: $v$8 for some constant $v$9 and every graph $$C_{\varphi}(v)=\{\varphi(v)\}\cup\{\varphi(vw):w\in N(v)\},$$0 (Coker et al., 2010). Combined with the Molloy–Reed upper bound on total chromatic number, this yields a universal degree bound

$$C_{\varphi}(v)=\{\varphi(v)\}\cup\{\varphi(vw):w\in N(v)\},$$1

for some absolute constant $$C_{\varphi}(v)=\{\varphi(v)\}\cup\{\varphi(vw):w\in N(v)\},$$2 (Coker et al., 2010). Thus the parameter is asymptotically of the form $$C_{\varphi}(v)=\{\varphi(v)\}\cup\{\varphi(vw):w\in N(v)\},$$3, rather than $$C_{\varphi}(v)=\{\varphi(v)\}\cup\{\varphi(vw):w\in N(v)\},$$4.

The constants obtained by the proof are deliberately non-optimized. The paper tracks them to show that, for sufficiently large $$C_{\varphi}(v)=\{\varphi(v)\}\cup\{\varphi(vw):w\in N(v)\},$$5,

$$C_{\varphi}(v)=\{\varphi(v)\}\cup\{\varphi(vw):w\in N(v)\},$$6

and, using Molloy–Reed,

$$C_{\varphi}(v)=\{\varphi(v)\}\cup\{\varphi(vw):w\in N(v)\},$$7

for $$C_{\varphi}(v)=\{\varphi(v)\}\cup\{\varphi(vw):w\in N(v)\},$$8 (Coker et al., 2010). The paper explicitly emphasizes that these values are not remotely optimal; their role is to prove the existence of an absolute additive constant.

A much sharper conjecture, due to Zhang et al. and recalled in (Coker et al., 2010), is

$$C_{\varphi}(v)=\{\varphi(v)\}\cup\{\varphi(vw):w\in N(v)\},$$9

for every graph $C_{\varphi}(u)\neq C_{\varphi}(v)$0 (Coker et al., 2010). This would be best possible: for odd complete graphs,

$C_{\varphi}(u)\neq C_{\varphi}(v)$1

when $C_{\varphi}(u)\neq C_{\varphi}(v)$2 is odd (Coker et al., 2010). The conjecture therefore predicts an exact universal additive constant.

3. Relation to total coloring and edge-distinguishing variants

The relation to ordinary total coloring is particularly tight. The inequality

$C_{\varphi}(u)\neq C_{\varphi}(v)$3

shows that the extra requirement of distinguishing adjacent vertices by incident color-sets raises the total chromatic number by at most a bounded additive term independent of the graph (Coker et al., 2010). A plausible implication is that progress on total coloring bounds immediately transfers to this parameter: if $C_{\varphi}(u)\neq C_{\varphi}(v)$4 were improved uniformly, the same would hold for $C_{\varphi}(u)\neq C_{\varphi}(v)$5.

The closest edge-only analogue is the adjacent vertex distinguishing edge chromatic number $C_{\varphi}(u)\neq C_{\varphi}(v)$6, in which only edges are colored and adjacent vertices must receive different sets of incident edge-colors (Coker et al., 2010). Hatami proved that

$C_{\varphi}(u)\neq C_{\varphi}(v)$7

for all graphs with $C_{\varphi}(u)\neq C_{\varphi}(v)$8, and (Coker et al., 2010) explicitly models its probabilistic strategy on Hatami’s argument (Coker et al., 2010). In this sense, $C_{\varphi}(u)\neq C_{\varphi}(v)$9 is the total-coloring analogue of the edge parameter $uv\in E(G)$0.

For sufficiently large $uv\in E(G)$1, the comparison chain becomes

$uv\in E(G)$2

for some absolute constant $uv\in E(G)$3 (Coker et al., 2010). Compared with the trivial non-complete, non-odd-cycle bound $uv\in E(G)$4, this places the parameter within a constant of the best possible lower bound $uv\in E(G)$5 (Coker et al., 2010).

4. Proof strategy and probabilistic machinery

The proof in (Coker et al., 2010) starts from an arbitrary proper total $uv\in E(G)$6-coloring and modifies it using only constantly many new colors. The vertex set is split into

$uv\in E(G)$7

where $uv\in E(G)$8, and the two parts are treated separately (Coker et al., 2010).

For low-degree vertices, the argument is deterministic. One keeps all edge colors and all high-degree vertex colors fixed, and repeatedly recolors a low-degree vertex $uv\in E(G)$9 whenever it is not distinguished from some neighbor (Coker et al., 2010). The crucial counting fact is that $\chi_{at}(G)$0 has at most $\chi_{at}(G)$1 forbidden colors, while a proper total coloring already uses at least $\chi_{at}(G)$2 colors, so at least one legal recoloring exists (Coker et al., 2010). This yields a proper total coloring in which every vertex of $\chi_{at}(G)$3 is distinguished from all neighbors.

For high-degree vertices, the proof is probabilistic and technically deeper. A random subset of edges is first selected and partially deleted to form a bounded-degree subgraph $\chi_{at}(G)$4; this makes most adjacent high-degree vertices differ substantially in their color-sets, while leaving only a controlled exceptional set (Coker et al., 2010). A second carefully chosen bounded set $\chi_{at}(G)$5 is then added to separate the remaining problematic pairs (Coker et al., 2010). The principal tools are Chernoff-type bounds, McDiarmid–Reed’s version of Talagrand’s inequality, and the symmetric Lovász Local Lemma (Coker et al., 2010).

Once $\chi_{at}(G)$6 has bounded maximum degree, Vizing’s theorem colors this subgraph with at most $\chi_{at}(G)$7 fresh colors, disjoint from the old palette, and this finishes the recoloring (Coker et al., 2010). The overall proof is existential, not algorithmic: the paper gives no polynomial-time construction and does not attempt derandomization (Coker et al., 2010).

5. Special cases, exact values, and related parameters

The paper (Coker et al., 2010) reports that Zhang et al. determined exact values of $\chi_{at}(G)$8 for cycles, complete graphs, complete bipartite graphs, and trees. It also records that for graphs with maximum degree $\chi_{at}(G)$9, Wang, Chen, and Hulgan independently proved

$G=(V,E)$0

and Hulgan further showed that such a coloring can be chosen so that at most one color appears on both edges and vertices (Coker et al., 2010). Before the general constant bound, Liu, An, and Gao had proved that if $G=(V,E)$1 is sufficiently large and

$G=(V,E)$2

then

$G=(V,E)$3

a result later subsumed by the removal of the minimum-degree hypothesis in (Coker et al., 2010).

At the same time, the phrase total distinguishing chromatic number is not completely uniform across the literature. A plausible implication is that several nearby notions coexist, all combining total colorings with some form of distinction. The following table summarizes the main related parameters that appear in the supplied literature.

Parameter	Distinguishing mechanism	Representative fact
$G=(V,E)$4	Adjacent vertices have different incident color-sets $G=(V,E)$5	$G=(V,E)$6 (Coker et al., 2010)
$G=(V,E)$7	Total coloring preserved only by the identity automorphism	$G=(V,E)$8 and $G=(V,E)$9 (Banerjee et al., 2024)
$\varphi:V\cup E\to [k]$0	Proper total coloring preserved only by the identity automorphism	$\varphi:V\cup E\to [k]$1 for connected infinite graphs (Imrich et al., 2019)
$\varphi:V\cup E\to [k]$2	Adjacent vertices have different sums $\varphi:V\cup E\to [k]$3	$\varphi:V\cup E\to [k]$4 (Loeb et al., 2015)
$\varphi:V\cup E\to [k]$5	Vertices at distance at most $\varphi:V\cup E\to [k]$6 have different total neighborhood color-sets	$\varphi:V\cup E\to [k]$7 (Wen et al., 2018)
$\varphi:V\cup E\to [k]$8	Proper total labeling with $\varphi:V\cup E\to [k]$9	Exact values are given for paths, cycles, stars, wheels, gears, and helms (Rohatgi et al., 2019)

Among these, $k$0 is the parameter explicitly identified with “total distinguishing chromatic number” in (Coker et al., 2010), whereas $k$1 and $k$2 are automorphism-based total variants, and $k$3 and $k$4 are stronger or differently structured distinguishing refinements (Banerjee et al., 2024, Imrich et al., 2019, Loeb et al., 2015, Wen et al., 2018).

6. Algorithmic status and mathematical significance

The existence theory for $k$5 is substantially stronger than the current algorithmic theory. The paper (Coker et al., 2010) does not discuss the computational complexity of computing $k$6, gives no explicit construction algorithm for the asserted bounds, and relies on probabilistic existence arguments that do not directly yield efficient procedures (Coker et al., 2010). Thus the main achievement is structural rather than algorithmic.

Its conceptual significance is twofold. First, it shows that distinguishing adjacent vertices by incident total color-sets is asymptotically no harder than total coloring itself: $k$7 Second, it places this parameter within a broader “$k$8” paradigm that also appears in adjacent-vertex-distinguishing edge colorings, neighbor-sum distinguishing total colorings, and radius-based strongly distinguishing total colorings [(Coker et al., 2010); (Loeb et al., 2015); (Wen et al., 2018)].

The main open direction remains the gap between the proved constant bound and the conjectured optimum $k$9 (Coker et al., 2010). A plausible implication is that progress on sharper total-coloring bounds, stronger versions of the Total Coloring Conjecture, or algorithmic forms of Local Lemma methods would all bear directly on the long-term structure of the total distinguishing chromatic number.