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

1,2 ,† 2,† 2,† Leandro M. Zatesko ∗ , Renato Carmo and Andre´ L. P. Guedes 1Federal University of Fronteira Sul, Chapeco,´ Brazil 2Department of Informatics, Federal University of Parana,´ 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)). 1 ∗ 2 1 ∪ 2 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 ) i | i| i  i for i 1,2 and  ∆,χ,χ0,χ00 ): χ00(G) 6 max n1,n2 + 1 + P(G1,G2) if G is a join graph, wherein ∈ { } ∈ { } { } B B P(G1,G2) := min ∆1 +∆2 +1,max χ10 ,χ200 ; χ00(G) 6 max n1,n2 +2(max ∆1 ,∆2 +1) if G is cobipartite, B { { }} { } { } wherein ∆i := maxu Vi dG[∂G(Vi)](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(G1,G2) is replaced by ∆1 + ∆2 + 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 The join of two graphs G1 = (V1,E1) and G2 = (V ,E ), denoted by G G , is the graph defined by 2 2 1 ∗ 2 Many variants of graph colouring problems have V(G1 G2) := V1 V2 and E(G1 G2) := E1 E2 v v ∗: v V and∪v V .A join∗ graph is∪ the re-∪ been developed and studied in the last century, each { 1 2 1 ∈ 1 2 ∈ 2} with its importance, applications, and open ques- sult of the join of two graphs. Remark that, if G1 tions3. Although most of these combinatorial op- and G2 are not the same K1 graph, we can assume timisation problems are NP-hard, there are some without loss of generality that they are disjoint (Zorzi polynomial-time algorithms which compute feasible and Zatesko, 2016). A cobipartite graph is the com- colourings using upper bounds for the optimal num- plement of a bipartite graph. Since a graph is a join ber of colours. In the particular case of total colour- graph if and only if it is the K1 or its complement is ings, which have applications e.g. in scheduling and disconnected, join graphs and cobipartite graphs can in task management in networks (Leidner, 2012), be recognised in linear time using, for instance, the some upper bounds for the total chromatic number of algorithms presented in (Ito and Yokoyama, 1998). a general n-order graph G of maximum degree ∆ are: χ (G) n + 1 (Behzad et al., 1967); • 00 6 χ (G) χ (G) + 2 χ(G) (Hind, 1990); • 00 6 0 χ (G) ∆ + 1026 (Molloyp and Reed, 1998); • 00 6 8 χ00(G) 6 ∆ + 8(ln∆) (Hind et al., 2000). Figure 1: In the left, the join graph K3 C4. In the right, a • cobipartite graph with n = 3 and n = ∗4. This paper presents potentially better upper bounds 1 2 for the case wherein G is a join or a cobipartite graph. Join graphs and cobipartite graphs have already

∗Partially supported by UFFS, 23205.001243/2016-30. been studied by several works in the context of edge- †Partially supported by CNPq, 428941/2016-8. colourings, with some partial results been found (Si- 3For an introduction on Graph Colouring we refer the mone and de Mello, 2006; Simone and Galluccio, reader to (Jensen and Toft, 1994). 2007; Simone and Galluccio, 2009; Machado and

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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 Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems

de Figueiredo, 2010; Simone and Galluccio, 2013; Vizing’s Recolouring Procedure. Also, it implies that Lima et al., 2015; Zorzi and Zatesko, 2016; Za- χ0(G) is either ∆(G) or ∆(G) + 1, in which case G is tesko et al., 2017). One of the reasons why hard said to be Class 1 or Class 2, respectively. Although graph problems are studied when restricted to join Vizing’s Recolouring Procedure yields a polynomial- graphs, for example, is the straightforward observa- time algorithm for computing a (∆(G) + 1)-edge- tion that the class of the join graphs includes some colouring of any graph G, deciding if G is Class 1 is other very important graph classes, such as graphs NP-complete (Holyer, 1981), even restricted to per- with a spanning star, complete multipartite graphs, fect graphs (Cai and Ellis, 1991), a class of graphs for and connected cographs. It is worthy mentioning that which optimal vertex-colourings can be computed in almost every graph can be turned into a cograph with polynomial time using linear programming — see, for n 1 1 1 2 example, (Grotschel¨ et al., 1988, Chapter 9). no more than 2 2 5 ηn edge addition or deletion operations (Corneil− et al., 1985; Alon and Stav, 2008). If G is a graph on n vertices with maximum degree This paper is structured
as follows. The remain- ∆ > n/3, the Overfull Graph Conjecture (Chetwynd ing of this section introduces some of the definitions and Hilton, 1984; Chetwynd and Hilton, 1986; Hilton used and some known facts relevant for the results we and Johnson, 1987) states that G is Class 2 if and present. In Sections 2 and 3, we present the upper only if it satisfies a property known as subgraph- bounds obtained for the total chromatic number of co- overfullness, which can be tested in polynomial time bipartite graphs and join graphs, respectively. In Sec- (Padberg and Rao, 1982; Niessen, 1994; Niessen, tion 4, we discuss how to improve the bound for some 2001). Join graphs and connected cobipartite graphs join graphs, and this improvement enlarges the class satisfy ∆ > n/2 by definition, but no polynomial-time of join graphs for which the Total Colouring Conjec- algorithm is known for computing the chromatic in- ture is known to be true. Finally, Section 5 concludes dex of all join or cobipartite graphs. Recall that all with remarks for future works. bipartite graphs are Class 1 (Konig,˝ 1916). Except for complete graphs and odd cycles, which Other Definitions and Related Results have χ(G) = ∆ + 1, χ(G) 6 ∆ by Brooks’s Theorem (Brooks, 1941). Therefore, χ00(G) 6 2∆ + 2. The In this paper, we use the term graph to refer always to Total Colouring Conjecture, proposed independently a simple graph, i.e. an undirected loopless graph with- by (Behzad, 1965) and (Vizing, 1968), states that out multiple edges. Definitions concerning to graph- χ00(G) 6 ∆ + 2 for every graph G. In view of that theoretical concepts follow their usual meanings and χ00(G) > ∆ + 1 by definition, graphs with χ00(G) = notation. Particularly, the degree of a vertex u in a ∆ + 1 and χ00(G) = ∆ + 2 have been called Type 1 and Type 2, respectively. This conjecture was proved for graph G is denoted by dG(u) := NG(u) = ∂G(u) , | | | | some graph classes, such as the complete graphs and wherein NG(u) and ∂G(u) denote, respectively, the set of the neighbours of u in G and the set of the edges the complete bipartite graphs (Behzad et al., 1967), and graphs with ∆ 3 n (Hilton and Hind, 1993). In incident to u in G. Also, for any X V(G), ∂G(X) > 4 denotes the cut defined by X in G, i.e.⊆ the set of the particular, the complete graph Kn is Type 1 if n is odd, edges of G with exactly one endpoint in X. or Type 2 otherwise, and the complete bipartite graph K is Type 1 if n = n , or Type 2 otherwise (Be- Let G be a graph and C be a set of t colours. A n1,n2 1 6 2 t-vertex-colouring is a function ϕ: V(G) C injec- hzad et al., 1967). Recall that computing the total tive in u,v for all uv E(G). The least→t for which chromatic number of a graph is NP-hard (Sanchez-´ G is t-vertex-colourable{ } ∈ is the chromatic number of Arroyo, 1989), even if restricted to bipartite graphs G, denoted by χ(G).A t-edge-colouring is a function (McDiarmid and Sanchez-Arroyo,´ 1994). ϕ: E(G) C injective in ∂ (u) for all u V(G). The A pullback from a graph G1 to a graph G2 is a → G ∈ least t for which G is t-edge-colourable is the chro- homeomorphism f : V(G1) V(G2) (i.e. a function → matic index of G, denoted by (G).A t-total colour- such that f (u) f (v) E(G2) for all uv E(G1)) in- χ0 ∈ ∈ ing is a function ϕ: V(G) E(G) C injective in jective in NG1 (u) u for all u V(G1). If there is ∪ → ∪ { } ∈ u,v and injective in ∂ (u) u for all u V(G) a pullback from G1 to G2, then χ0(G1) 6 χ0(G2) and { } G ∪ { } ∈ and all v N (u). The least t for which G is t-total χ00(G1) 6 χ00(G2) (de Figueiredo et al., 1999). ∈ G colourable is the total chromatic number of G, de- Now, let C be a set of colours, no matter how noted by χ00(G). Obviously, χ00(G) 6 χ(G) + χ0(G). many. Under an assignment of a list L(u) C for If uv E(G), G uv is t-edge-colourable, and each u V(G), a vertex-list-colouring is a⊆ vertex- ∈ − ∈ dG uv(w) < t for all w NG(u) u , then G is also colouring ϕ: V(G) C such that ϕ(u) L(u) for t-edge-colourable− (Vizing,∈ 1964).∪ { Vizing’s} proof for all u V(G). The→ graph G is said to be∈ t-vertex- this statement is constructive and often referred as choosable∈ if it is vertex-list-colourable under any

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

assignment of lists to the vertices with at least t Step 3. Now, for each v V , the set X(v) of the ∈ 2 colours in each list. The least t for which G is t- colours assigned to the neighbours of v in BG vertex-choosable is the vertex-choosability of G, de- and to the edges incident to v in BG has at most noted by ch(G). Analogously, under assignments 2dBG (v) colours. Hence, if we take the list L(v) := of lists to the edges, we define edge-list-colourings, C X(v), we have \ and the least t for which G is t-edge-choosable B B L(v) > max n1,n2 + 2(∆1 + 1) 2∆2 > n2. is the edge-choosability of G, denoted by ch0(G). | | { } − Since ch(Kn ) = n2 is a straightforward result, we Clearly, ch(G) > χ(G), ch0(G) > χ0(G), and χ00(G) 6 2 can assign to each v V2 a colour of L(v). ch0(G) + 2. ∈ The Edge-List-Colouring Conjecture4 states that Step 4. Finally, in order to complete ϕ, it remains to colour the edges of E(G[V ]) E(G[V ]). For ch0(G) = χ0(G) for every graph G. It is worthy 1 ∪ 2 remarking that the similar statement concerning to each uv amongst them, let X(uv) be the set of vertex-list-colourings is known to be false, since one the colours assigned to the vertices u and v and can construct a graph with χ(G) = 2 and ch(G) ar- to the edges of BG adjacent to uv in G. Define bitrarily large (Gravier, 1996), although it is true that then the list L(uv) := C X(uv). Since X(uv) 6 2∆B + 2, L(uv) max\ n ,n . Thus,| by| the ch(G) 6 ∆+1 (Vizing, 1976; Erdos˝ et al., 1979). The 1 | | > { 1 2} Edge-List-Colouring Conjecture has been shown only result of (Haggkvist¨ and Janssen, 1997) accord- for a few graphs, such as the bipartite graphs (Janssen, ing to which ch0(Kn) 6 n, we can assign to each uv E(G[V ]) E(G[V ]) a colour of L(uv). 1993; Galvin, 1995) and the Kn with n odd (Haggkvist¨ ∈ 1 ∪ 2 and Janssen, 1997) or n 1 prime (Schauz, 2014). Because all the colourings taken in the proof of For the K with n even and−n 1 composite, it is only Theorem 1 can be obtained in polynomial time, our n − known that ch0(Kn) 6 ∆(Kn) + 1 = n (Haggkvist¨ and proof is a polynomial-time algorithm to construct a Janssen, 1997). Observe that the K is Class 1 if and (max n ,n +2(max ∆B,∆B +1))-total colouring. n { 1 2} { 1 2 } only if n is even, a standard result which can be found Recall that ∆(G) = max n 1 + ∆B,n 1 + ∆B , { 1 − 1 2 − 2 } e.g. in (Fiorini and Wilson, 1977). which means that the upper bound provided in The- orem 1 is better than the bounds for general graphs B B listed in Section 1, as long as ∆1 and ∆2 are not too 2 COBIPARTITE GRAPHS large, in the sense that the propositions below clarify. Proposition 2. If

Throughout this section, G is a connected cobipar- B B min n1,n2 tite graph with V(G) = V V , wherein V and V max ∆1 ,∆2 6 { } 1, 1 ∪ 2 1 2 { } 2 − are two disjoint cliques with V =: n and V =: n . B B 1 1 2 2 then max n1,n2 + 2(max ∆ ,∆ + 1) is strictly Connectivity is assumed without| | loss of generality| | be- { } { 1 2 } less than V(G) + 1, the upper bound for χ00(G) by cause the total chromatic number of a graph is the (Behzad et| al., 1967).| maximum amongst the total chromatic numbers of its B B Proof. max n1,n2 + 2(max ∆ ,∆ + 1) connected components. Ergo, ∂G(V1) = ∂G(V2) = 0/, { } { 1 2 } 6 max n ,n + min n ,n and we define BG := G[∂G(V1)] = G[∂G(V2)]. Note 6 { 1 2} { 1 2} that B is bipartite. = n1 + n2 < V(G) + 1. G | | B Theorem 1. Let ∆i := maxu Vi dBG (u) for i 1,2 . Proposition 3. If Then, χ (G) max n ,n +∈ 2(max ∆B,∆B∈+ { 1).} B B 25 00 6 { 1 2} { 1 2 } max ∆1 ,∆2 6 5 10 2, B { } ×B B − Proof. Let C be a set with max n1,n2 + 2(∆1 + 1) then max n1,n2 + 2(max ∆ ,∆ + 1) is strictly { } B { } { 1 2 } colours, assuming without loss of generality that ∆1 > less than ∆(G) + 1026, the upper bound for χ (G) by B 00 ∆2 . We shall construct a total colouring ϕ for G using (Molloy and Reed, 1998). the colours of C . B B Proof. max n1,n2 + 2(max ∆1 ,∆2 + 1) Step 1. Choose n1 colours from C and assign each { } B { } B 26 6 max n1 1 + ∆1 ,n2 1 + ∆2 + 10 2 one of them to a vertex of V1. { − 26 − } − Step 2. For each uv E(B ) with u V and v V , < ∆(G) + 10 ∈ G ∈ 1 ∈ 2 create the list L(uv) with any ∆(BG) colours of Proposition 4. If C distinct from ϕ(u). As ∆(BG) = ch0(BG), by B B 3 max ∆1 ,∆2 6 max n1,n2 , (Galvin, 1995), we can assign to each uv a colour { } { } − 2 of L(uv). then max n ,n + 2(maxp ∆B,∆B + 1) is strictly { 1 2} { 1 2 } 4For more on the origin and the history of this conjec- less than χ0(G) + 2 χ(G), the upper bound for ture, see (Jensen and Toft, 1994, Chapter 12). χ (G) by (Hind, 1990). 00 p

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

B B Proof. max n1,n2 + 2(max ∆1 ,∆2 + 1) be computed in polynomial time, then our proof is a { } { } polynomial-time algorithm, provided that the under- 6 ∆(G) + 2 max n1,n2 1 { } − lying colourings are also known or can be computed. < ∆(G) + 2 χ(G) χ0(G) + 2 χ(G). p 6 Replacing P(G1,G2) by some upper bound on it, such Proposition 5. If p p as ∆1 +∆2 +1, also makes our algorithm polynomial. Similar to the bound for the cobipartite graphs, the B B B B 8 3 upper bound presented in Theorem 6 is better than max ∆1 ,∆2 6 4 ln(max n1 + ∆1 ,n2 + ∆2 ) { } { } − 2 the upper bounds for general graphs listed in Section
then max n ,n + 2(max ∆B,∆B + 1) is strictly 1 if P(G1,G2) is not too large, in the sense that the { 1 2} { 1 2 } less than ∆(G) + 8(ln∆(G))8, the upper bound for propositions below clarify. χ00(G) by (Hind et al., 2000). Proposition 7. If P(G ,G ) min n ,n 1, then 1 2 6 { 1 2} − Proof. max n ,n + 2(max ∆B,∆B + 1) max n1,n2 + 1 + P(G1,G2) < V(G) + 1. { 1 2} { 1 2 } { } | | (G) + ( ( (G)))8 Proof. max n1,n2 + 1 + P(G1,G2) 6 ∆ 8 ln ∆ 1. { } − max n ,n + min n ,n = V(G) . 6 { 1 2} { 1 2} | | 26 Proposition 8. If P(G1,G2) 6 10 1, then −26 3 JOIN GRAPHS max n1,n2 + 1 + P(G1,G2) < ∆(G) + 10 . { } Proof. max n1,n2 + 1 + P(G1,G2) Throughout this section and the next, G is the join { } 26 < max n1 + ∆2,n2 + ∆1 + 10 . of two disjoint graphs G1 and G2 with, respectively, { } n1 and n2 vertices and maximum degrees ∆1 and ∆2. Proposition 9. If P(G ,G ) 2√χ + χ 1, then 1 2 6 1 2 − Now, BG denotes the complete bipartite graph G max n ,n + 1 + P(G ,G ) < χ (G) + 2 χ(G). − { 1 2} 1 2 0 (E1 E2). For simplicity, we write χ1 := χ(G1), χ20 := ∪ Proof. max n1,n2 + 1 + P(G1,G2) p χ0(G2) etc. Note that ∆(G) = max ∆1 + n2,∆2 + n1 . { } { } < max n + ∆ ,n + ∆ + 2√χ + χ Theorem 6. Let { 1 2 2 1} 1 2 6 χ0(G) + 2 χ(G). P(G1,G2) := min ∆1 + ∆2 + 1,max χ10 ,χ200 . (1) { { }} Proposition 10. If p Then, χ (G) max n ,n + 1 + P(G ,G ). 00 6 1 2 1 2 8 { } P(G ,G ) 8 ln(max n + ∆B,n + ∆B ) 1, Proof. Let t := max n ,n +1+P(G ,G ) and take 1 2 6 { 1 1 2 2 } − { 1 2} 1 2 two disjoint sets CA and CB with, respectively, χ1 then max n1,n2 + 1 + P(G1,G2) is strictly less
than { } 8 and max χ10 ,χ200 colours. As it can be straightfor- ∆(G) + 8(ln∆(G)) . wardly verified{ that} C + C t, take a set C with A B 6 Proof. max n ,n + 1 + P(G ,G ) t colours having C |and|C | as| subsets. We shall con- 1 2 1 2 A B { } 8 struct a total colouring ϕ: V(G) E(G) C . < max n1 + ∆2,n2 + ∆1 + 8(ln∆(G)) . ∪ → { } Step 1. Take a χ1-vertex-colouring of G1 using only the colours of CA. 4 IMPROVING THE BOUND FOR Step 2. Take a χ10 -edge-colouring of G1 and a χ200- total colouring of G2, both using only the colours JOIN GRAPHS of CB. Since CA and CB are disjoint, no colour conflict has been created. Following (Simone and de Mello, 2006), we denote Step 3. Now, for each edge uv B , with u V , let by GM the graph (G1 G2)+M for any perfect match- ∈ G ∈ 1 ∪ X(uv) be the set of the colours assigned to the ver- ing M on BG. Inspired by an observation in the same tices u and v and to the edges of G G adjacent work, we show how the upper bound of Theorem 6 1 ∪ 2 to uv in G. It is clear that X(uv) 6 1+P(G1,G2). may be lowered in some cases. In the statements, as Define then the list L(uv| ) := C| X(uv). Since it is usual for functions f : A B, we denote by f (X) \ ( ) → L(uv) > t 1 P(G1,G2) = max n1,n2 and the set x X f x for all X A. | | − − { } ∈ ⊆ ch0(BG) = max n1,n2 (Galvin, 1995), we can as- TheoremS 11. Let ϕ be a total colouring of GM for sign to each uv{ E(B}) a colour of L(uv). ∈ G some perfect matching M on BG. If the sets ϕ(V1) and ϕ(E M V E ) are disjoint and Remark in Theorem 6 that, from the definition of 1 ∪ ∪ 2 ∪ 2 P(G1,G2) in (1), the choice of the graphs for the roles ϕ(E1 M V2 E2) 6 max χ10 ,χ200 6 ∆1 + ∆2 + 3, of G1 or G2 may lead to a better or a worse upper | ∪ ∪ ∪ | { } bound. Moreover, if P(G ,G ) is known, or if it can then χ (G) max n ,n + max χ ,χ . 1 2 00 6 { 1 2} { 10 200}

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

Proof. Let C be a set with t := max n1,n2 + yields a (max χ300,∆3 + 2 )-total colouring ϕ of GM max χ ,χ colours having C := ϕ(V ) {and C }:= with ϕ(V ) and{ϕ(E M } V E ) disjoint and { 10 200} A 1 B 1 1 ∪ ∪ 2 ∪ 2 ϕ(E1 M V2 E2) as subsets. In order to obtain a ∪ ∪ ∪ ϕ(E M V E ) max χ00,∆ + 2 t-total colouring of G using the colours of C , we start | 1 ∪ ∪ 2 ∪ 2 | 6 { 3 3 } with the total colouring ϕ of GM, remaining to colour 6 ∆1 + ∆2 + 3. only the edges of B M. G − The rest of the proof follows as the proof for We proceed now as in Step 3 of the proof of The- Theorem 11, but with max χ300,∆3 + 2 instead of orem 6. For each edge uv BG M, with u V1, let { } max χ0 ,χ00 . X(uv) be the set of the colours∈ assigned− to the∈ vertices { 1 2} u and v and to the edges of GM adjacent to uv in G. Corollary 14. If there is a graph G3 satisfying the Clearly preconditions of Theorem 13, and if max n1,n2 + max χ ,∆ + 2 max n + ∆ ,n + ∆ {+ 2, then} { 300 3 } 6 { 1 2 2 1} X(uv) 1 + min ∆1 + ∆2 + 3,max χ0 ,χ00 the Total Colouring Conjecture is true for G. | | 6 { { 1 2}} = 1 + max χ0 ,χ00 . Theorem 15 and Corollary 16 below deal with { 1 2} the joins of unitary interval graphs. Unitary interval Therefore, if we define the list L(uv) := C X(uv), graphs are also known as indifference graphs, whose we have L(uv) = max n ,n 1 = ∆(B )\. Since | | { 1 2} − G edge-colourings have been studied by (de Figueiredo ∆(BG) = ch0(BG) (Galvin, 1995), we can assign to et al., 1997; de Figueiredo et al., 2000; de Figueiredo each uv E(B ) a colour of L(uv). ∈ G et al., 2003), with some partial results been found. Corollary 12. If G has a total colouring ϕ of Theorem 15. If G1 and G2 are indifference graphs, G , for some perfect matching M on B , sat- M G then χ00(G) 6 max n1,n2 + max ∆1,∆2 + 2. isfying the preconditions of Theorem 11, and if { } { } Proof. The proof follows from Theorem 13 by taking max n1,n2 + max χ10 ,χ200 6 max n1 + ∆2 + 2,n2 + { } { } { G3 := Kmax , +1. By (de Figueiredo et al., 1997), ∆1 + 2 , then the Total Colouring Conjecture is true ∆1 ∆2 } there is a pullback{ } from any indifference graph D on for G, i.e. χ00(G) 6 ∆(G) + 2. k vertices to the K` for any ` > k (de Figueiredo et al., Theorem 13. If there is a graph G3 such that 1999), and, moreover, if 0,...,k 1 is an indifference order of D, a pullback can be given− by the function 1. max χ00,∆3 + 2 6 ∆1 + ∆2 + 3, { 3 } f (i) = i mod `, under V(K ) = 0,...,` 1 . There- 2. there are a pullback f from G to G and a pull- ` 13 1 3 fore, back to our join graph G, it{ is clear− that} a match- back f23 from G2 to G3, and ing M on BG satisfying the requirements of Theorem 3. there is a perfect matching M on BG such that, for 13 can be taken. all uv M with u V1, f13(u) = f23(v), ∈ ∈ Corollary 16. If G1 and G2 are indifference graphs, then χ (G) max n ,n + max χ ,∆ + 2 . 00 6 { 1 2} { 300 3 } and if n1 = n2 or ∆1 = ∆2, then the Total Colouring Conjecture is true for G. Proof. Let CA be a set with χ1 colours and take any optimal vertex-colouring of G1. Let CB be a set with max χ ,∆ + 2 colours, disjoint from C , and ψ be { 300 3 } A a total colouring of G3 using the colours of CB. By 5 FUTURE WORKS (de Figueiredo et al., 1999), the function ϕ : E C 1 1 → B defined by In order to obtain a total colouring, the algorithms presented in the proofs of Theorems 1 and 6 decom- ϕ(uv) = ψ( f (u) f (v)), uv E , 13 13 ∀ ∈ 1 pose the input graph and, considering the parts of the decomposition in an appropriate order, work with the is a proper edge-colouring of G1, as the function solutions for other colouring problems for each part. ϕ2 : V2 E2 CB defined by ∪ → All these problems are hard combinatorial optimisa- ϕ(u) = ψ( f23(u)), u V2, tion problems, and we hope further investigation on ∀ ∈ them considering the restricted cases of the graphs ob- ϕ(uv) = ψ( f23(u) f23(v)), uv E2, ∀ ∈ tained by our decompositions can lead to better upper is a proper total colouring of G2. Since it is clear that bounds for the total chromatic number of join and co- max χ ,∆ + 2 > ∆ + 1, at least one colour α bipartite graphs. We also encourage future works to { 300 3 } 3 x ∈ CB is missing at each x V3, i.e. αx is not the colour investigate other decompositions. assigned by ψ to x nor to∈ any edge incident to x. Ergo, Theorem 15 and Corollary 16 form an example of for all uv M with u V1, the colour α f (u) is missing how Theorem 13 can be applied in order to prove the at both u and∈ v and thence∈ can be assigned to uv. This Total Colouring Conjecture for a noteworthy subclass

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

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