On-Line List Colouring of Graphs

This paper studies on-line list colouring of graphs. It is proved that the on-line choice number of a graph $G$ on $n$ vertices is at most $\chi(G) \ln n+1$, and the on-line $b$-choice number of $G$ is at most ${e\chi(G)-1\over e-1} (b-1+ \ln n)+b$. Suppose $G$ is a graph with a given $\chi(G)$-colouring of $G$. Then for any $(\chi(G) \ln n +1)$-assignment $L$ of $G$, we give a polynomial time algorithm which constructs an $L$-colouring of $G$. For any $({e\chi(G)-1\over e-1} (b-1+ \ln n)+b)$-assignment $L$ of $G$, we give a polynomial time algorithm which constructs an $(L,b)$-colouring of $G$. We then characterize all on-line $2$-choosable graphs. It is also proved that a complete bipartite graph of the form $K_{3,q}$ is on-line $3$-choosable if and only if it is $3$-choosable, but there are graphs of the form $K_{6,q}$ which are $3$-choosable but not on-line $3$-choosable. Some open questions concerning on-line list colouring are posed in the last section.

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Chip Games and Paintability

The Electronic Journal of Combinatorics ◽

10.37236/5723 ◽

2016 ◽

Vol 23 (3) ◽

Cited By ~ 3

Author(s):

Lech Duraj ◽

Grzegorz Gutowski ◽

Jakub Kozik

Keyword(s):

Bipartite Graph ◽

Complete Bipartite Graph ◽

Uniform Hypergraph ◽

Natural Family ◽

Uniform Hypergraphs ◽

Classical Correspondence ◽

On Line ◽

Choice Number ◽

The Difference

We prove that the paint number of the complete bipartite graph $K_{N,N}$ is $\log N + O(1)$. As a consequence, we get that the difference between the paint number and the choice number of $K_{N,N}$ is $\Theta(\log \log N)$. This answers in the negative the question of Zhu (2009) whether this difference, for all graphs, can be bounded by a common constant. By a classical correspondence, our result translates to the framework of on-line coloring of uniform hypergraphs. This way we obtain that for every on-line two coloring algorithm there exists a $k$-uniform hypergraph with $\Theta(2^k)$ edges on which the strategy fails. The results are derived through an analysis of a natural family of chip games.

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Learning Importance of Preferences

10.29007/v68w ◽

2018 ◽

Author(s):

Ying Zhu ◽

Mirek Truszczynski

Keyword(s):

Polynomial Time ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Scoring Rules ◽

Np Hard ◽

Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

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