Generalized list colouring of graphs
Eun-Kyung Cho, Ilkyoo Choi, Yiting Jiang, Ringi Kim, Boram Park, Jiayan Yan, Xuding Zhu
aa r X i v : . [ m a t h . C O ] F e b Generalized list colouring of graphs
Eun-Kyung Cho ∗ , Ilkyoo Choi † , Yiting Jiang ‡ , Ringi Kim § , Boram Park ¶ , Jiayan Yan k , Xuding Zhu ∗∗ February 20, 2020
Abstract
This paper disproves a conjecture in [ Wang, Wu, Yan and Xie, AWeaker Version of a Conjecture on List Vertex Arboricity of Graphs,Graphs and Combinatorics (2015) 31:17791787] and answers in nega-tive a question in [ Dvoˇr´ak, Pek´arek and Sereni, On generalized choiceand coloring numbers, arXiv: 1081.0682403, 2019]. In return, we posefive open problems.
Keywords: generalized list colouring, list vertex arboricity, list star arboric-ity, choice number.
Assume G is a hereditary family of graphs, i.e., if G ∈ G and H is an inducedsubgraph of G , then H ∈ G . A G -colouring of a graph G is a colouring φ of the vertices of G so that each colour class induces a graph in G . A G - n -colouring of G is a G -colouring φ of G such that φ ( v ) ∈ [ n ] = { , , . . . , n } for each vertex v . We say G is G - n -colourable if there exists a G - n -colouringof G . The G -chromatic number of G is χ G ( G ) = min { n : G is G - n -colourable } . ∗ Department of Mathematics, Hankuk University of Foreign Studies, Yongin-si,Gyeonggi-do, Republic of Korea. [email protected] † Department of Mathematics, Hankuk University of Foreign Studies, Yongin-si,Gyeonggi-do, Republic of Korea. [email protected] ‡ Department of Mathematics, Zhejiang Normal University, China. [email protected] § Department of Mathematical Sciences, KAIST, Daejeon, Republic of Korea. [email protected] ¶ Department of Mathematics, Ajou University, Suwon-si, Gyeonggi-do, Republic ofKorea. [email protected] k Department of Mathematics, Zhejiang Normal University, China. [email protected] ∗∗ Department of Mathematics, Zhejiang Normal University, China. [email protected] L is a list assignment of G . A G - L -colouring of G is a G -colouring φ of G so that φ ( v ) ∈ L ( v ) for each vertex v . We say G is G - n -choosable iffor every n -list assignment L of G , there exists a G - L -colouring of G , The G -choice number of G is ch G ( G ) = min { n : G is G - n -choosable } . The concept of G -colouring of a graph is a slight modification of theconcept of generalized colouring of graphs introduced in [1], where the graphclass G is assumed to be of the form G = { G : f ( G ) ≤ d } for some graphparameter f and constant d . We find that there are some graph families G for which the G -colouring problems are interesting, and yet G is not easilyexpressed in such a form.Many colouring concepts studied in the literature are G -colourings forspecial graph families G .We denote by • G k the family of graphs whose connected components are of order atmost k ; • D k the family of graphs of maximum degree at most k ; • F the family of forests; • S the family of star forests; • L the family of linear forests; • C k the family of graphs of colouring number at most k . • M k the family of graphs of maximum average degree at most k .Many of the G -colourings have special names and are studied extensivelyin the literature. • A G k -colouring of G is a colouring of G with clustering k . In particular,a G -colouring of G is a proper colouring of G . • A D k -colouring of G is a k -defective colouring of G . The parameter χ D k ( G ) is the k -defective chromatic number of G . Also, a D -colouringof G is a proper colouring of G . • An F -colouring of G is a vertex arboreal colouring of G . The param-eter χ F ( G ) is the vertex arboricity of G , and ch F ( G ) is the list vertexarboricity of G . • The parameter χ S ( G ) is the star vertex arboricity of G , and ch S ( G ) isthe star list vertex arboricity of G .2 The parameter χ L ( G ) is the linear vertex arboricity of G , and ch L ( G )is the linear list vertex arboricity of G .For any two graph families G and G ′ , for any graph G , it follows easilyfrom the definition that χ G ( G ) ≤ (max H ∈ G ′ χ G ( H )) χ G ′ ( G ) , (1)and this upper bound is tight. For example, χ ( G ) ≤ χ F ( G ) , χ S ( G ) ≤ χ F ( G ) and χ ( G ) ≤ ( k + 1) χ M k ( G ) , and for any integers k, k ′ , χ G k ( G ) ≤ (cid:24) k ′ k (cid:25) χ G k ′ ( G ) , and equalities hold for some graphs G .It is natural to ask if the same or similar inequalities hold for the corre-sponding choice number. Some of such inequalities are posed as conjecturesor questions in the literature. For example, the following conjecture wasproposed in [2]: Conjecture 1.1
For any graph G , ch ( G ) ≤ ch F ( G ) . The following question was asked in [1]:
Question 1.2
Is it true that for any graph G , for any positive integer k , ch ( G ) ≤ ( k + 1) ch M k ( G )?In this note, we disprove Conjecture 1.1 and give a negative answer toQuestion 1.2. Lemma 2.1
Assume k ≥ and m = k ( k + 1) − are integers. Then forany positive integer n , ch S ( K m,n ) ≤ k .Proof. Assume k, n ≥ m = k ( k + 1) −
1. Let G = K m,n be the complete bipartite graph with partite sets A, B , with | A | = m and B = n . We show that ch S ( G ) ≤ k .Let L be a k -list assignment of G . Build a bipartite graph H with partitesets A and C = ∪ v ∈ A L ( v ), and in which vc is an edge if and only if c ∈ L ( v ).Note that each vertex v ∈ A has degree k in H .3 subset C ′ of C is heavy if | N H ( C ′ ) | ≥ ( k + 1) | C ′ | . In particular, ∅ is aheavy subset of C . Let C ′ be a maximal heavy subset of C . Let A ′ = N H ( C ′ )and H ′ = H − ( A ′ ∪ C ′ ).Then each vertex v ∈ A − A ′ has degree k in H ′ . If there is a colour c for which d H ′ ( c ) ≥ k + 1, then let C ′′ = C ′ ∪ { c } . Then | N H ( C ′′ ) | = | N H ( C ′ ) | + d H ′ ( c ) ≥ ( k + 1) | C ′′ | . So C ′′ is heavy, contraryto our assumption that C ′ is a maximum heavy subset of C .So each vertex c ∈ C − C ′ has degree at most k in H ′ . By Hall’s Theorem,there is a matching M in H ′ that covers all the vertices of A − A ′ . Let φ bethe L -colouring of A − A ′ defined as φ ( v ) = c if vc ∈ M . So all vertices of A − A ′ are coloured by distinct colours. Extend φ to an L -colouring of H as follows: • Since k ( k + 1) > | A | ≥ | N H ( C ′ ) | ≥ | C ′ | ( k + 1), we know that | C ′ | ≤ k −
1. For each vertex v ∈ B , we have L ( v ) − C ′ = ∅ . Let φ ( v ) be anycolour in L ( v ) − C ′ . • For each vertex v ∈ A ′ , as A ′ = N H ( C ′ ), L ( v ) ∩ C ′ = ∅ . Let φ ( v ) beany colour in L ( v ) ∩ C ′ .This is an S - L -colouring of G , as each connected monochromatic sub-graph of G contains at most one vertex of A , and hence is a star. Thiscompletes the proof of Lemma 2.1. ✷ It is well-known that if n ≥ m m , then ch ( K m,n ) = m + 1. The fol-lowing lemma shows that for any constant d , if n is sufficiently large, then ch D d ( K m,n ) = m + 1. Lemma 2.2
Assume d is a non-negative integer. If n ≥ ( dm + 1) m m , then ch D d ( K m,n ) = m + 1 .Proof. Assume n ≥ ( dm + 1) m m and G = K m,n with partite sets A, B ,where | A | = m and | B | = n . As G is m -degenerate, we have ch D d ( G ) ≤ ch ( G ) ≤ m + 1.Now we show that ch D d ( G ) > m .Let L be the m -assignment which assigns to vertices v ∈ A pairwisedisjoint m -sets { L ( v ) : v ∈ A } . Let Φ be the set of all L -colourings φ of A . Thus | Φ | = m m . For each φ ∈ Φ, assign a ( dm + 1)-subset B φ of B so that for distinct φ, φ ′ ∈ Φ, B φ ∩ B φ ′ = ∅ . Since | B | ≥ ( dm + 1) m m ,such an assignment exists. Extend L to an m -assignment of G by letting L ( v ) = φ ( A ) for any v ∈ B φ . Assign arbitrary m colours to v if v ∈ B is notcontained in any subsets B φ .Now we show that G is not D d - L -colourable. Assume to the contrarythat φ is a D d - L -colouring of G . Let φ | A be the restriction of φ to A . For4ny v ∈ B φ | A , φ ( v ) ∈ L ( v ) = φ ( A ). As | φ ( A ) | = m and | B φ | A | = ( dm + 1),there exists a colour c ∈ φ ( A ) such that | φ − ( c ) ∩ B φ | A | ≥ d + 1. Assume u ∈ A and c = φ ( u ). Then u has at least d + 1 neighbours that are colouredthe same colour as u itself. So φ is not a D d - L -colouring of G .This completes the proof of Lemma 2.2. ✷ As a corollary of Lemmas 2.1 and 2.2, we have the following theorem.
Theorem 2.3
For any integers k, d with k ≥ , there exists a graph G with ch S ( G ) ≤ k and ch D d ( G ) = k ( k + 1) . In particular, for any constant p , thereexists a graph G with ch ( G ) ≥ p · ch S ( G ) ≥ p · ch F ( G ) ≥ p · ch M ( G ) . This theorem refutes Conjecture 1.1 and gives a negative answer to Ques-tion 1.2. We remark that Conjecture 1.1, posed at the end of [2], is not theconjecture referred to in the title of that paper. The main conjecture studiedin [2] is the following conjecture posed in [3]:
Conjecture 2.4 If | V ( G ) | ≤ χ F ( G ) , then ch F ( G ) = χ F ( G ) . This conjecture remains open.It is known [1] that ch ( G ) is bounded from above by a function of ch G ( G ), provided that graphs in G have bounded maximum average degree.Or equivalently, graphs in G have bounded choice number. In particular, ch ( G ) ≤ f ( ch F ( G )) for some function f . The function f found in [1] isexponential. Theorem 2.3 shows that f cannot be a linear function. Itwould be interesting to know if there is a polynomial function f such that ch ( G ) ≤ f ( ch F ( G )). Question 2.5
Are there constant integers a, b such that ch ( G ) ≤ a ( ch F ( G )) b ? If so, what is the smallest such integer b ? It would also be interesting to know if the bound given in Theorem 2.3is tight. I.e., is it true that ch ( G ) ≤ ch F ( G )( ch F ( G ) + 1) for all graphs G ?As observed in the introduction, for any two graph classes G and G ′ , χ G ( G ) ≤ (max H ∈ G ′ χ G ( H )) χ G ′ ( G ). We are interested in the question whetherthe same inequality holds for the corresponding choice number. If G ′ ⊆ G ,then trivially, the inequality ch G ( G ) ≤ (max H ∈ G ′ ch G ( H )) ch G ′ ( G ) = ch G ′ ( G )holds. We do now know any non-trivial case where the inequality ch G ( G ) ≤ (max H ∈ G ′ ch G ( H )) ch G ′ ( G ) holds. As remarked in [1], the following questionmay have a positive answer. 5 uestion 2.6 [1] Is it true that for any graph G and any positive integer k , ch ( G ) ≤ kch G k ( G )?Even the k = 2 case of the above question is very interesting and chal-lenging. More generally, the following question seems to be natural andinteresting: Question 2.7
Is it true that for any graph G and any positive integers k, k ′ , ch G k ′ ( G ) ≤ (cid:24) kk ′ (cid:25) ch G k ( G )?The relation between ch L ( G ) and ch S ( G ) is also interesting. By Theorem2.3, there are graphs G for which ch L ( G ) ≥ ch D ( G ) ≥ ch S ( G )( ch S ( G ) + 1) . It follows from (1) that χ S ( G ) ≤ χ L ( G ) . The following questions remain open.
Question 2.8
Is it true that for any graph G , ch S ( G ) ≤ ch L ( G )? Question 2.9
Is it true that for any graph G , ch L ( G ) ≤ ch S ( G )( ch S ( G ) + 1)? Or is there an integer a such that ch L ( G ) ≤ ( ch S ( G )) a ? References [1] Z. Dvoˇr´ak, J. Pek´arek and J. Sereni,
On generalized choice and coloringnumbers , arXiv: 1081.0682403, 2019.[2] W. Wang, B. Wu, Z. Yan and N. Xie,
A Weaker Version of a Conjectureon List Vertex Arboricity of Graphs . Graphs and Combinatorics (2015)31:17791787 DOI 10.1007/s00373-014-1466-5[3] L. Zhen and B. Wu.