Upper Bounds for the Total Chromatic Number of Join Graphs and

Cobipartite Graphs

Leandro M. Zatesko

1,2∗,†

, Renato Carmo

2,†

and Andr

e L. P. Guedes

2,†

Federal University of Fronteira Sul, Chapec

o, Brazil

Department of Informatics, Federal University of Paran

a, Curitiba, Brazil

Keywords:

Combinatorial Optimisation, Total Colouring, Join Graphs, Cobipartite Graphs.

Abstract:

We concern ourselves with the combinatorial optimisation problem of determining a minimum total colouring

of a graph G for the case wherein G is a join graph G = G

∗G

or a cobipartite graph G = (V

∪V

,E(G)).

We present algorithms for computing a feasible, not necessarily optimal, solution for this problem, providing

the following upper bounds for the total chromatic numbers of these graphs (let n

= |V

| and 

= (G

)

for i ∈ {1,2} and  ∈ {∆,χ, χ

,χ

}): χ

(G) 6 max{n

}+ 1 + P(G

) if G is a join graph, wherein

P(G

)

= min{∆

+∆

+1,max{χ

,χ

}}; χ

(G) 6 max{n

}+2(max{∆

,∆

}+1) if G is cobipartite,

wherein ∆

= max

u∈V

G[∂

)]

(u) for i ∈{1, 2}. Our algorithm for the cobipartite graphs runs in polynomial

time. Our algorithm for the join graphs runs in polynomial time if P(G

) is replaced by ∆

+ ∆

+ 1 or if

it can be computed in polynomial time. We also prove the Total Colouring Conjecture for some subclasses of

join graphs, such as some joins of indifference (unitary interval) graphs.

1 INTRODUCTION

Many variants of graph colouring problems have

been developed and studied in the last century, each

with its importance, applications, and open ques-

tions

. Although most of these combinatorial op-

timisation problems are NP-hard, there are some

polynomial-time algorithms which compute feasible

colourings using upper bounds for the optimal num-

ber of colours. In the particular case of total colour-

ings, which have applications e.g. in scheduling and

in task management in networks (Leidner, 2012),

some upper bounds for the total chromatic number of

a general n-order graph G of maximum degree ∆ are:

• χ

(G) 6 n + 1 (Behzad et al., 1967);

• χ

(G) 6 χ

(G) + 2

χ(G) (Hind, 1990);

• χ

(G) 6 ∆ + 10

(Molloy and Reed, 1998);

• χ

(G) 6 ∆ + 8(ln ∆)

(Hind et al., 2000).

This paper presents potentially better upper bounds

for the case wherein G is a join or a cobipartite graph.

∗

Partially supported by UFFS, 23205.001243/2016-30.

†

Partially supported by CNPq, 428941/2016-8.

For an introduction on Graph Colouring we refer the

reader to (Jensen and Toft, 1994).

The join of two graphs G

= (V

) and G

), denoted by G

∗G

, is the graph deﬁned by

V (G

∗G

)

= V

∪V

and E(G

∗G

)

= E

∪E

∪

: v

∈V

and v

∈V

}. A join graph is the re-

sult of the join of two graphs. Remark that, if G

and G

are not the same K

graph, we can assume

without loss of generality that they are disjoint (Zorzi

and Zatesko, 2016). A cobipartite graph is the com-

plement of a bipartite graph. Since a graph is a join

graph if and only if it is the K

or its complement is

disconnected, join graphs and cobipartite graphs can

be recognised in linear time using, for instance, the

algorithms presented in (Ito and Yokoyama, 1998).

Figure 1: In the left, the join graph K

∗C

. In the right, a

cobipartite graph with n

= 3 and n

= 4.

Join graphs and cobipartite graphs have already

been studied by several works in the context of edge-

colourings, with some partial results been found (Si-

mone and de Mello, 2006; Simone and Galluccio,

2007; Simone and Galluccio, 2009; Machado and

Zatesko, L., Carmo, R. and Guedes, A.

Upper Bounds for the Total Chromatic Number of Join Graphs and Cobipartite Graphs.

DOI: 10.5220/0006627102470253

In Proceedings of the 7th International Conference on Operations Research and Enterprise Systems (ICORES 2018), pages 247-253

ISBN: 978-989-758-285-1

247

de Figueiredo, 2010; Simone and Galluccio, 2013;

Lima et al., 2015; Zorzi and Zatesko, 2016; Za-

tesko et al., 2017). One of the reasons why hard

graph problems are studied when restricted to join

graphs, for example, is the straightforward observa-

tion that the class of the join graphs includes some

other very important graph classes, such as graphs

with a spanning star, complete multipartite graphs,

and connected cographs. It is worthy mentioning that

almost every graph can be turned into a cograph with

no more than





−

ηn

edge addition or deletion

operations (Corneil et al., 1985; Alon and Stav, 2008).

This paper is structured as follows. The remain-

ing of this section introduces some of the deﬁnitions

used and some known facts relevant for the results we

present. In Sections 2 and 3, we present the upper

bounds obtained for the total chromatic number of co-

bipartite graphs and join graphs, respectively. In Sec-

tion 4, we discuss how to improve the bound for some

join graphs, and this improvement enlarges the class

of join graphs for which the Total Colouring Conjec-

ture is known to be true. Finally, Section 5 concludes

with remarks for future works.

Other Deﬁnitions and Related Results

In this paper, we use the term graph to refer always to

a simple graph, i.e. an undirected loopless graph with-

out multiple edges. Deﬁnitions concerning to graph-

theoretical concepts follow their usual meanings and

notation. Particularly, the degree of a vertex u in a

graph G is denoted by d

(u)

= |N

(u)| = |∂

(u)|,

wherein N

(u) and ∂

(u) denote, respectively, the set

of the neighbours of u in G and the set of the edges

incident to u in G. Also, for any X ⊆ V (G), ∂

(X)

denotes the cut deﬁned by X in G, i.e. the set of the

edges of G with exactly one endpoint in X.

Let G be a graph and C be a set of t colours. A

t-vertex-colouring is a function ϕ : V (G) → C injec-

tive in {u,v} for all uv ∈ E(G). The least t for which

G is t-vertex-colourable is the chromatic number of

G, denoted by χ(G). A t-edge-colouring is a function

ϕ: E(G) →C injective in ∂

(u) for all u ∈V (G). The

least t for which G is t-edge-colourable is the chro-

matic index of G, denoted by χ

(G). A t-total colour-

ing is a function ϕ: V (G) ∪E(G) → C injective in

{u,v} and injective in ∂

(u) ∪{u} for all u ∈ V (G)

and all v ∈ N

(u). The least t for which G is t-total

colourable is the total chromatic number of G, de-

noted by χ

(G). Obviously, χ

(G) 6 χ(G) + χ

(G).

If uv ∈ E(G), G − uv is t-edge-colourable, and

G−uv

(w) < t for all w ∈ N

(u) ∪{u}, then G is also

t-edge-colourable (Vizing, 1964). Vizing’s proof for

this statement is constructive and often referred as

Vizing’s Recolouring Procedure. Also, it implies that

(G) is either ∆(G) or ∆(G) + 1, in which case G is

said to be Class 1 or Class 2, respectively. Although

Vizing’s Recolouring Procedure yields a polynomial-

time algorithm for computing a (∆(G) + 1)-edge-

colouring of any graph G, deciding if G is Class 1 is

NP-complete (Holyer, 1981), even restricted to per-

fect graphs (Cai and Ellis, 1991), a class of graphs for

which optimal vertex-colourings can be computed in

polynomial time using linear programming — see, for

example, (Gr

otschel et al., 1988, Chapter 9).

If G is a graph on n vertices with maximum degree

∆ > n/3, the Overfull Graph Conjecture (Chetwynd

and Hilton, 1984; Chetwynd and Hilton, 1986; Hilton

and Johnson, 1987) states that G is Class 2 if and

only if it satisﬁes a property known as subgraph-

overfullness, which can be tested in polynomial time

(Padberg and Rao, 1982; Niessen, 1994; Niessen,

2001). Join graphs and connected cobipartite graphs

satisfy ∆ > n/2 by deﬁnition, but no polynomial-time

algorithm is known for computing the chromatic in-

dex of all join or cobipartite graphs. Recall that all

bipartite graphs are Class 1 (K

onig, 1916).

Except for complete graphs and odd cycles, which

have χ(G) = ∆ + 1, χ(G) 6 ∆ by Brooks’s Theorem

(Brooks, 1941). Therefore, χ

(G) 6 2∆ + 2. The

Total Colouring Conjecture, proposed independently

by (Behzad, 1965) and (Vizing, 1968), states that

(G) 6 ∆ + 2 for every graph G. In view of that

(G) > ∆ + 1 by deﬁnition, graphs with χ

(G) =

∆ + 1 and χ

(G) = ∆ + 2 have been called Type 1 and

Type 2, respectively. This conjecture was proved for

some graph classes, such as the complete graphs and

the complete bipartite graphs (Behzad et al., 1967),

and graphs with ∆ >

n (Hilton and Hind, 1993). In

particular, the complete graph K

is Type 1 if n is odd,

or Type 2 otherwise, and the complete bipartite graph

is Type 1 if n

6= n

, or Type 2 otherwise (Be-

hzad et al., 1967). Recall that computing the total

chromatic number of a graph is NP-hard (S

anchez-

Arroyo, 1989), even if restricted to bipartite graphs

(McDiarmid and S

anchez-Arroyo, 1994).

A pullback from a graph G

to a graph G

is a

homeomorphism f : V (G

) →V (G

) (i.e. a function

such that f (u) f (v) ∈ E(G

) for all uv ∈ E(G

)) in-

jective in N

(u) ∪{u} for all u ∈V (G

). If there is

a pullback from G

to G

, then χ

) 6 χ

) and

) 6 χ

) (de Figueiredo et al., 1999).

Now, let C be a set of colours, no matter how

many. Under an assignment of a list L(u) ⊆ C for

each u ∈ V (G), a vertex-list-colouring is a vertex-

colouring ϕ: V (G) → C such that ϕ(u) ∈ L(u) for

all u ∈ V (G). The graph G is said to be t-vertex-

choosable if it is vertex-list-colourable under any

ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems

248

assignment of lists to the vertices with at least t

colours in each list. The least t for which G is t-

vertex-choosable is the vertex-choosability of G, de-

noted by ch(G). Analogously, under assignments

of lists to the edges, we deﬁne edge-list-colourings,

and the least t for which G is t-edge-choosable

is the edge-choosability of G, denoted by ch

(G).

Clearly, ch(G) > χ(G), ch

(G) > χ

(G), and χ

(G) 6

(G) + 2.

The Edge-List-Colouring Conjecture

states that

(G) = χ

(G) for every graph G. It is worthy

remarking that the similar statement concerning to

vertex-list-colourings is known to be false, since one

can construct a graph with χ(G) = 2 and ch(G) ar-

bitrarily large (Gravier, 1996), although it is true that

ch(G) 6 ∆ +1 (Vizing, 1976; Erd

os et al., 1979). The

Edge-List-Colouring Conjecture has been shown only

for a few graphs, such as the bipartite graphs (Janssen,

1993; Galvin, 1995) and the K

with n odd (H

aggkvist

and Janssen, 1997) or n −1 prime (Schauz, 2014).

For the K

with n even and n −1 composite, it is only

known that ch

) 6 ∆(K

) + 1 = n (H

aggkvist and

Janssen, 1997). Observe that the K

is Class 1 if and

only if n is even, a standard result which can be found

e.g. in (Fiorini and Wilson, 1977).

2 COBIPARTITE GRAPHS

Throughout this section, G is a connected cobipar-

tite graph with V (G) = V

∪V

, wherein V

and V

are two disjoint cliques with |V

and |V

Connectivity is assumed without loss of generality be-

cause the total chromatic number of a graph is the

maximum amongst the total chromatic numbers of its

connected components. Ergo, ∂

) = ∂

) 6=

and we deﬁne B

= G[∂

)] = G[∂

)]. Note

that B

is bipartite.

Theorem 1. Let ∆

= max

u∈V

(u) for i ∈ {1,2}.

Then, χ

(G) 6 max{n

}+ 2(max{∆

,∆

}+ 1).

Proof. Let C be a set with max{n

}+ 2(∆

+ 1)

colours, assuming without loss of generality that ∆

∆

. We shall construct a total colouring ϕ for G using

the colours of C .

Step 1. Choose n

colours from C and assign each

one of them to a vertex of V

Step 2. For each uv ∈ E(B

) with u ∈V

and v ∈V

create the list L(uv) with any ∆(B

) colours of

C distinct from ϕ(u). As ∆(B

) = ch

), by

(Galvin, 1995), we can assign to each uv a colour

of L(uv).

For more on the origin and the history of this conjec-

ture, see (Jensen and Toft, 1994, Chapter 12).

Step 3. Now, for each v ∈ V

, the set X(v) of the

colours assigned to the neighbours of v in B

and to the edges incident to v in B

has at most

(v) colours. Hence, if we take the list L(v)

C \X(v), we have

|L(v)| > max{n

}+ 2(∆

+ 1) −2∆

> n

Since ch(K

) = n

is a straightforward result, we

can assign to each v ∈V

a colour of L(v).

Step 4. Finally, in order to complete ϕ, it remains

to colour the edges of E(G[V

]) ∪E(G[V

]). For

each uv amongst them, let X(uv) be the set of

the colours assigned to the vertices u and v and

to the edges of B

adjacent to uv in G. Deﬁne

then the list L(uv)

= C \X(uv). Since |X (uv)| 6

2∆

+ 2, |L(uv)| > max{n

}. Thus, by the

result of (H

aggkvist and Janssen, 1997) accord-

ing to which ch

) 6 n, we can assign to each

uv ∈ E(G[V

]) ∪E(G[V

]) a colour of L(uv).

Because all the colourings taken in the proof of

Theorem 1 can be obtained in polynomial time, our

proof is a polynomial-time algorithm to construct a

(max{n

}+2(max{∆

,∆

}+1))-total colouring.

Recall that ∆(G) = max{n

−1 + ∆

which means that the upper bound provided in The-

orem 1 is better than the bounds for general graphs

listed in Section 1, as long as ∆

and ∆

are not too

large, in the sense that the propositions below clarify.

Proposition 2. If

max{∆

,∆

} 6

min{n

}

−1,

then max{n

} + 2(max{∆

,∆

} + 1) is strictly

less than |V (G)|+ 1, the upper bound for χ

(G) by

(Behzad et al., 1967).

Proof. max{n

}+ 2(max{∆

,∆

}+ 1)

6 max{n

}+ min{n

}

= n

+ n

< |V (G)|+1.

Proposition 3. If

max{∆

,∆

} 6 5 ×10

−2,

then max{n

} + 2(max{∆

,∆

} + 1) is strictly

less than ∆(G) + 10

, the upper bound for χ

(G) by

(Molloy and Reed, 1998).

Proof. max{n

}+ 2(max{∆

,∆

}+ 1)

6 max{n

−1 + ∆

}+ 10

−2

< ∆(G) + 10

Proposition 4. If

max{∆

,∆

} 6

max{n

}−

then max{n

} + 2(max{∆

,∆

} + 1) is strictly

less than χ

(G) + 2

χ(G), the upper bound for

(G) by (Hind, 1990).

Upper Bounds for the Total Chromatic Number of Join Graphs and Cobipartite Graphs

249

Proof. max{n

}+ 2(max{∆

,∆

}+ 1)

6 ∆(G) + 2

max{n

}−1

< ∆(G) + 2

χ(G) 6 χ

(G) + 2

χ(G).

Proposition 5. If

max{∆

,∆

} 6 4



ln(max{n

+ ∆

})



−

then max{n

} + 2(max{∆

,∆

} + 1) is strictly

less than ∆(G) + 8(ln ∆(G))

, the upper bound for

(G) by (Hind et al., 2000).

Proof. max{n

}+ 2(max{∆

,∆

}+ 1)

6 ∆(G) + 8(ln(∆(G)))

−1.

3 JOIN GRAPHS

Throughout this section and the next, G is the join

of two disjoint graphs G

and G

with, respectively,

and n

vertices and maximum degrees ∆

and ∆

Now, B

denotes the complete bipartite graph G −

∪E

). For simplicity, we write χ

= χ(G

), χ

) etc. Note that ∆(G) = max{∆

+ n

,∆

+ n

Theorem 6. Let

P(G

)

= min{∆

+ ∆

+ 1,max{χ

,χ

}}. (1)

Then, χ

(G) 6 max{n

}+ 1 + P(G

Proof. Let t

= max{n

}+1 +P(G

) and take

two disjoint sets C

and C

with, respectively, χ

and max{χ

,χ

} colours. As it can be straightfor-

wardly veriﬁed that |C

|+ |C

| 6 t, take a set C with

t colours having C

and C

as subsets. We shall con-

struct a total colouring ϕ: V (G) ∪E(G) → C .

Step 1. Take a χ

-vertex-colouring of G

using only

the colours of C

Step 2. Take a χ

-edge-colouring of G

and a χ

total colouring of G

, both using only the colours

of C

. Since C

and C

are disjoint, no colour

conﬂict has been created.

Step 3. Now, for each edge uv ∈ B

, with u ∈V

, let

X(uv) be the set of the colours assigned to the ver-

tices u and v and to the edges of G

∪G

adjacent

to uv in G. It is clear that |X(uv)|6 1+P(G

Deﬁne then the list L(uv)

= C \X(uv). Since

|L(uv)| > t − 1 − P(G

) = max{n

} and

) = max{n

}(Galvin, 1995), we can as-

sign to each uv ∈ E(B

) a colour of L(uv).

Remark in Theorem 6 that, from the deﬁnition of

P(G

) in (1), the choice of the graphs for the roles

of G

or G

may lead to a better or a worse upper

bound. Moreover, if P(G

) is known, or if it can

be computed in polynomial time, then our proof is a

polynomial-time algorithm, provided that the under-

lying colourings are also known or can be computed.

Replacing P(G

) by some upper bound on it, such

as ∆

+ ∆

+ 1, also makes our algorithm polynomial.

Similar to the bound for the cobipartite graphs, the

upper bound presented in Theorem 6 is better than

the upper bounds for general graphs listed in Section

1 if P(G

) is not too large, in the sense that the

propositions below clarify.

Proposition 7. If P(G

) 6 min{n

}−1, then

max{n

}+ 1 + P(G

) < |V (G)|+ 1.

Proof. max{n

}+ 1 + P(G

)

6 max{n

}+ min{n

} = |V (G)|.

Proposition 8. If P(G

) 6 10

− 1, then

max{n

}+ 1 + P(G

) < ∆(G) + 10

Proof. max{n

}+ 1 + P(G

)

< max{n

+ ∆

}+ 10

Proposition 9. If P(G

) 6 2

√

+ χ

−1, then

max{n

}+ 1 + P(G

) < χ

(G) + 2

χ(G).

Proof. max{n

}+ 1 + P(G

)

< max{n

+ ∆

}+ 2

√

+ χ

6 χ

(G) + 2

χ(G).

Proposition 10. If

P(G

) 6 8



ln(max{n

+ ∆

})



−1,

then max{n

}+ 1 + P(G

) is strictly less than

∆(G) + 8(ln ∆(G))

Proof. max{n

}+ 1 + P(G

)

< max{n

+ ∆

}+ 8(ln ∆(G))

4 IMPROVING THE BOUND FOR

JOIN GRAPHS

Following (Simone and de Mello, 2006), we denote

by G

the graph (G

∪G

)+M for any perfect match-

ing M on B

. Inspired by an observation in the same

work, we show how the upper bound of Theorem 6

may be lowered in some cases. In the statements, as

it is usual for functions f : A → B, we denote by f (X)

the set

x∈X

f (x) for all X ⊆ A.

Theorem 11. Let ϕ be a total colouring of G

for

some perfect matching M on B

. If the sets ϕ(V

) and

ϕ(E

∪M ∪V

∪E

) are disjoint and

|ϕ(E

∪M ∪V

∪E

)| 6 max{χ

,χ

} 6 ∆

+ ∆

+ 3,

then χ

(G) 6 max{n

}+ max{χ

,χ

ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems

250

Proof. Let C be a set with t

= max{n

} +

max{χ

,χ

} colours having C

= ϕ(V

) and C

ϕ(E

∪M ∪V

∪E

) as subsets. In order to obtain a

t-total colouring of G using the colours of C , we start

with the total colouring ϕ of G

, remaining to colour

only the edges of B

−M.

We proceed now as in Step 3 of the proof of The-

orem 6. For each edge uv ∈ B

−M, with u ∈V

, let

X(uv) be the set of the colours assigned to the vertices

u and v and to the edges of G

adjacent to uv in G.

Clearly

|X(uv)| 6 1 + min{∆

+ ∆

+ 3,max{χ

,χ

}}

= 1 + max{χ

,χ

Therefore, if we deﬁne the list L(uv)

= C \X(uv),

we have |L(uv)| = max{n

}−1 = ∆(B

). Since

∆(B

) = ch

) (Galvin, 1995), we can assign to

each uv ∈ E(B

) a colour of L(uv).

Corollary 12. If G has a total colouring ϕ of

, for some perfect matching M on B

, sat-

isfying the preconditions of Theorem 11, and if

max{n

}+ max{χ

,χ

} 6 max{n

+ ∆

+ 2,n

∆

+ 2}, then the Total Colouring Conjecture is true

for G, i.e. χ

(G) 6 ∆(G) + 2.

Theorem 13. If there is a graph G

such that

1. max{χ

,∆

+ 2} 6 ∆

+ ∆

+ 3,

2. there are a pullback f

from G

to G

and a pull-

back f

from G

to G

, and

3. there is a perfect matching M on B

such that, for

all uv ∈ M with u ∈V

, f

(u) = f

(v),

then χ

(G) 6 max{n

}+ max{χ

,∆

+ 2}.

Proof. Let C

be a set with χ

colours and take any

optimal vertex-colouring of G

. Let C

be a set with

max{χ

,∆

+ 2} colours, disjoint from C

, and ψ be

a total colouring of G

using the colours of C

. By

(de Figueiredo et al., 1999), the function ϕ

: E

→C

deﬁned by

ϕ(uv) = ψ( f

(u) f

(v)), ∀uv ∈ E

is a proper edge-colouring of G

, as the function

: V

∪E

→ C

deﬁned by

ϕ(u) = ψ( f

(u)), ∀u ∈V

ϕ(uv) = ψ( f

(u) f

(v)), ∀uv ∈ E

is a proper total colouring of G

. Since it is clear that

max{χ

,∆

+ 2} > ∆

+ 1, at least one colour α

∈

is missing at each x ∈V

, i.e. α

is not the colour

assigned by ψ to x nor to any edge incident to x. Ergo,

for all uv ∈ M with u ∈V

, the colour α

f (u)

is missing

at both u and v and thence can be assigned to uv. This

yields a (max{χ

,∆

+ 2})-total colouring ϕ of G

with ϕ(V

) and ϕ(E

∪M ∪V

∪E

) disjoint and

|ϕ(E

∪M ∪V

∪E

)| 6 max{χ

,∆

+ 2}

6 ∆

+ ∆

+ 3.

The rest of the proof follows as the proof for

Theorem 11, but with max{χ

,∆

+ 2} instead of

max{χ

,χ

Corollary 14. If there is a graph G

satisfying the

preconditions of Theorem 13, and if max{n

max{χ

,∆

+ 2} 6 max{n

+ ∆

}+ 2, then

the Total Colouring Conjecture is true for G.

Theorem 15 and Corollary 16 below deal with

the joins of unitary interval graphs. Unitary interval

graphs are also known as indifference graphs, whose

edge-colourings have been studied by (de Figueiredo

et al., 1997; de Figueiredo et al., 2000; de Figueiredo

et al., 2003), with some partial results been found.

Theorem 15. If G

and G

are indifference graphs,

then χ

(G) 6 max{n

}+ max{∆

,∆

}+ 2.

Proof. The proof follows from Theorem 13 by taking

= K

max{∆

,∆

}+1

. By (de Figueiredo et al., 1997),

there is a pullback from any indifference graph D on

k vertices to the K

for any ` > k (de Figueiredo et al.,

1999), and, moreover, if 0,. .. ,k −1 is an indifference

order of D, a pullback can be given by the function

f (i) = i mod `, under V (K

) = {0,...,` −1}. There-

fore, back to our join graph G, it is clear that a match-

ing M on B

satisfying the requirements of Theorem

13 can be taken.

Corollary 16. If G

and G

are indifference graphs,

and if n

= n

or ∆

= ∆

, then the Total Colouring

Conjecture is true for G.

5 FUTURE WORKS

In order to obtain a total colouring, the algorithms

presented in the proofs of Theorems 1 and 6 decom-

pose the input graph and, considering the parts of the

decomposition in an appropriate order, work with the

solutions for other colouring problems for each part.

All these problems are hard combinatorial optimisa-

tion problems, and we hope further investigation on

them considering the restricted cases of the graphs ob-

tained by our decompositions can lead to better upper

bounds for the total chromatic number of join and co-

bipartite graphs. We also encourage future works to

investigate other decompositions.

Theorem 15 and Corollary 16 form an example of

how Theorem 13 can be applied in order to prove the

Total Colouring Conjecture for a noteworthy subclass

Upper Bounds for the Total Chromatic Number of Join Graphs and Cobipartite Graphs

251

of join graphs. Similar results could also be obtained

for other subclasses, such as the joins of graphs with

no more than max{∆

,∆

}+ 1 per connected com-

ponent, since a pullback from each component to the

max{∆

,∆

}+1

could be easily obtained. We encourage

future works to investigate more applications.

ACKNOWLEDGEMENTS

We thank the anonymous referee for the reading and

the suggestions given.

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