On-line list colouring of graphs Xuding Zhu Zhejiang Normal University

Suppose G is a graph A list assignment L assigns to each vertex x a set L(x) of permissible colours. An L-colouring h of G assigns to each vertex x a colour such that for every edge xy choice number

Given a vertex x, L(x) tells us which colours are permissible. Alternately, given a colour i, one can ask which vertices have i as a permissible colour. A list assignment L can be given as Given a colour i, is the set of vertices having i as a permissible colour. An L-colouring is a family of independent subsets and

on-line f-list colouring game on G played by Alice and Bob At round i, Alice choose a set of uncoloured vertices. is the set of vertices which has colour i as a permissible colour. is the number of permissible colours for x Bob chooses an independent subset of and colour vertices in by colour i. Alice wins the game if there is a vertex x, which has been given f(x) permissible colours and remains uncoloured. Otherwise, eventually all vertices are coloured and Bob wins the game.

G is on-line f-choosable if Bob has a winning strategy for the on-line f-list colouring game. G is on-line k-choosable if G is on-line f-choosable for f(x)=k for every x. The on-line choice number of G is the minimum k for which G is on-line k-choosable.

Theorem [Erdos-Rubin-Taylor (1979)] is 2-choosable. is not on-line 2-choosable

Alice wins the game

Question: Can the difference be arbitrarily large ? Question: Can the ratio be arbitrarily large ?

Most upper bounds for choice number are also upper bounds for on-line choice number. Currently used method in proving upper bounds for choice number Kernel method InductionSome works for on-line choice number, Combinatorial Nullstellensatz Theorem [Schauz,2009] For planar G, Theorem [Chung-Z,2011] For planar G, triangle free + no 4-cycle adjacent to a 4-cycle or a 5-cycle, Theorem [Schauz,2009] Upper bounds for ch(G) proved by Combinatorial nullstellensatz works for on-line choice number

Theorem [Schauz,2009] Upper bounds for ch(G) proved by Combinatorial nullstellensatz works for on-line choice number Theorem [Schauz,2009] If G has an orientation D with then G is on-line The proof is by induction (no polynomial is involved).

Probabilistic methodDoes not work for on-line choice number Theorem [Alon, 1992] The proof is by probabilistic method Theorem[Z,2009] Proof: If G is bipartite and has n vertices, then

If a vertex x has permissible colours, Bob will be able to colour it. Bob colours, double the weight of each vertex in A B Initially, each vertex x has weight w(x)=1 Assume Alice has given set If The total weight of uncoloured vertices is not increased. If a vertex is given a permissible colour but is not coloured by that colour, then it weight doubles. If a vertex x has given k permissible colours, but remains uncoloured, then

A graph G is chromatic choosable if Conjecture: Line graphs are chromatic choosable. Conjecture: Claw-free graphs are chromatic choosable. Conjecture: Total graphs are chromatic choosable. Conjecture [Ohba] Graphs G with are chromatic choosable. Conjecture: For any G, for any k > 1, is chromatic choosable. Theorem [Noel-Reed-Wu]

On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable |33|333 NOT TRUE

On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable |33|333 Alice’s move

On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable |33|333 Alice’s move 3|23|233 23|23|33 Bob’s (2 possible) moves

On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable |33|333 Alice’s move 3|23|233 23|23|33 Bob’s (2 possible) moves 13|222 2|3|222

On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable |33|333 Alice’s move 3|23|233 23|23|33 Bob’s (2 possible) moves 13|222 2|3|222 3|111

On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable |33|333 Alice’s move 3|23|233 23|23|33 Bob’s (2 possible) moves 13|222 2|3|222 3|111 2|112

On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable |33|333 Alice’s move 3|23|233 23|23|33 13|222 2|3|222 3|111 2|112 3|13|22 2|3|11

Theorem [Kim-Kwon-Liu-Z,2012] For n > 1, is not on-line -choosable

On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable.

Theorem [Erdos-Rubin-Taylor (1979)] is chromatic choosable. Proof Assume each vertex is given n permissible colours. then colour them by a common colour use induction to colour the rest. If for some k, have a common permissible colour and

Assume no partite set has a common permissible colour Build a bipartite graph colours By Hall’s theorem, there is a matching that covers all the vertices of V V C The proof does not work for on-line list colouring

On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable. Question: is on-line n-choosable ?

Question:Theorem [Huang-Wong-Z,2010] is on-line n-choosable. Proof Combinatorial Nullstellensatz An explicit winning strategy for Bob ( Kim-Kwon-Liu-Z, 2012)

Theorem [Kim-Kwon-Liu-Z, 2012] G: complete multipartite graph with partite sets satisfying the following Then (G,f) is feasible, i.e., G is on-line f-choosable. is on-line n-choosable.

Alice’s choice Bob’s choice After this round, G changed and f changed. Need to prove:new (G,f) still satisfies the condition

Alice’s choice

Bob colours v

Alice’s choice

Bob colours v

Theorem [Kim-Kwon-Liu-Z, 2012] G: complete multipartite graph with partite sets satisfying the following Then (G,f) is feasible, i.e., G is on-line f-choosable. is on-line n-choosable.

Theorem [Kozik-Micek-Z, 2012] G: complete multipartite graph with partite sets satisfying the following Then (G,f) is feasible.

On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable. Conjecture holds for graphs with independence number 3

Open Problems Can the difference ch^{OL}-ch be arbitrarily big? On-line Ohba conjecture true?

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