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### Restricted list coloring and Hall's condition

M.M. Cropper, J.L. Goldwasser
2002 Discrete Mathematics
If L is a list assignment of colors to the vertices of a graph G of chromatic number (G), a certain condition on L and G, known as Hall's condition, which is obviously necessary for G to have an L-coloring  ...  But if the set of colors has size greater than (G), then Hall's condition is again su cient if and only if each block of G is a clique.  ...  For example, the graphs and list assignments shown in Fig. 1 satisfy Hall's condition, but there is no list coloring.  ...

### Completing Partial Proper Colorings using Hall's Condition

Sarah Holliday, Jennifer Vandenbussche, Erik E Westlund
2015 Electronic Journal of Combinatorics
In the context of list-coloring the vertices of a graph, Hall's condition is a generalization of Hall's Marriage Theorem and is necessary (but not sufficient) for a graph to admit a proper list-coloring  ...  A list assignment \$L\$ to a graph \$G\$ is called Hall if \$(G,L)\$ satisfies Hall's condition.  ...  Acknowledgements The authors thank SibelÖzkan and Peter Johnson for personal communication. The authors also thank the referees for their time and helpful comments.  ...

### Page 5471 of Mathematical Reviews Vol. , Issue 98I [page]

1998 Mathematical Reviews
Hall’s condition gave rise to two graph parameters, the Hall number and the Hall-condition number, which have been of some use in posing and attacking problems concerning list colorings.  ...  “In this paper we generalize Hall’s condition to a necessary condition for the existence of a proper list-multicoloring of the graph, from the given lists.  ...

### Extending Hall's Theorem into List Colorings: A Partial History

D. G. Hoffman, P. D. Johnson
2007 International Journal of Mathematics and Mathematical Sciences
Hilton and P. D. Johnson Jr. found a natural generalization of the condition in Philip Hall's celebrated theorem on systems of distinct representatives.  ...  New results are presented concerning list colorings of independence systems and colorings of graphs with nonnegative measurable functions on positive measure spaces.  ...  condition and graph colorings, 1989-2003 The content of Hall's theorem is that when G is complete, Hall's condition is sufficient for the existence of a proper L-coloring of G.  ...

### Author index

2002 Discrete Mathematics
Goldwasser, Restricted list coloring and Hall's condition (1-3) 57-63 Denley, T., Arithmetic progressions of cycles in outer-planar graphs (1-3) 65-70 Jamison, R.E., The Helly bound for singular sums (  ...  ., Hall's condition for list-coloring, and the Hall parameters: recent developments (1-3) 135-147 K .undgen, A., Minimum average distance subsets in the hamming cube  ...

### Page 3203 of Mathematical Reviews Vol. , Issue 2003e [page]

2003 Mathematical Reviews
Finally, he considers the concept of restricted list-coloring parameters as those for which the set of colors C satisfies |C| = constant and generalizes the Hall’s condition for coloring to multicoloring  ...  D., Jr. (1-ABRN-DS; Auburn, AL) Hall’s condition for list-coloring, and the Hall parameters: recent developments. Combinatorics, graph theory and computing (Louisville, KY, 1999).  ...

### Choosability of a weighted path and free-choosability of a cycle [article]

Yves Aubry
2011 arXiv   pre-print
In this paper, we give necessary and sufficient conditions for a weighted path to be (L,w)-colorable for some list assignments L. Furthermore, we solve the problem of the free-choosability of a cycle.  ...  A graph G with a list of colors L(v) and weight w(v) for each vertex v is (L,w)-colorable if one can choose a subset of w(v) colors from L(v) for each vertex v, such that adjacent vertices receive disjoint  ...  We say that G is (L, w)-colorable if there exists an (L, w)-coloring c of G.  ...

### Page 1593 of Mathematical Reviews Vol. , Issue 2003C [page]

2003 Mathematical Reviews
L. (1-WV; Morgantown, WV) Restricted list coloring and Hall’s condition. (English summary) Combinatorics, graph theory and computing (Louisville, KY, 1999). Discrete Math. 249 (2002), no. 1-3, 57-63.  ...  Summary: “If L is a list assignment of colors to the vertices of a graph G of chromatic number 7(G), a certain condition on L and G, known as Hall’s condition, which is obviously necessary for G to have  ...

### On choosability of some complete multipartite graphs and Ohba's conjecture

Yufa Shen, Wenjie He, Guoping Zheng, Yanning Wang, Lingmin Zhang
2008 Discrete Mathematics
These results have been obtained indirectly from the investigation about complete multipartite graphs by Gravier and Maffray and by Enomoto et al.  ...  It is clear that Ohba's conjecture is true if and only if it is true for complete multipartite graphs.  ...  Acknowledgments We would like to thank the referees for their careful reading and valuable comments. In particular, we are indebted to one of them for pointing out a flaw in the proof of Theorem 3.2.  ...

### Ohba's conjecture is true for graphs with independence number at most three

Yufa Shen, Wenjie He, Guoping Zheng, Yanpo Li
2009 Applied Mathematics Letters
In this work we show that if |V (G)| ≤ 2χ (G)+1 and the independence number of G is at most 3, then G is chromatic-choosable.  ...  This proves that Ohba's conjecture is true for all graphs G with independence number at most 3 and all χ (G)-chromatic subgraphs of G.  ...  Acknowledgements We would like to express our gratitude to the referees for their careful reading and valuable comments and suggestions regarding this work.  ...

### Beyond Ohba's Conjecture: A bound on the choice number of k-chromatic graphs with n vertices

Jonathan A. Noel, Douglas B. West, Hehui Wu, Xuding Zhu
2015 European journal of combinatorics (Print)
We thank Bruce Reed for pointing out a simplification of the proof, and we thank an anonymous referee for many useful suggestions concerning the exposition.  ...  A Sufficient Condition for L-Coloring By Corollary 2.5, each color appears in at most two lists in each part.  ...  To obtain such a set of colors (and thereby define the remaining merges), we again apply Hall's Theorem. Definition 5.2.  ...

### Beyond Ohba's Conjecture: A bound on the choice number of k-chromatic graphs with n vertices [article]

Jonathan A. Noel and Douglas B. West and Hehui Wu and Xuding Zhu
2014 arXiv   pre-print
Let ch(G) denote the choice number of a graph G (also called "list chromatic number" or "choosability" of G). Noel, Reed, and Wu proved the conjecture of Ohba that ch(G)=χ(G) when |V(G)|< 2χ(G)+1.  ...  Our result is sharp for |V(G)|< 3χ(G) using Ohba's examples, and it improves the best-known upper bound for ch(K_4,...,4).  ...  We thank Bruce Reed for pointing out a simplification of the proof, and we thank an anonymous referee for many useful suggestions concerning the exposition.  ...

### List colorings with measurable sets

Jan Hladký, Daniel Kráal, Jean-Sébastien Sereni, Michael Stiebitz
2008 Journal of Graph Theory
We provide a simpler proof of a measurable generalization of Hall's theorem due to Hilton and Johnson [J.  ...  Graph Theory 54 (2007), 179-193] and show that the measurable list chromatic number of a finite graph G is equal to its fractional chromatic number.  ...  The authors would like to thank their hosts for providing excellent environment for this research, and sharing their mathematical insights.  ...

### Page 4112 of Mathematical Reviews Vol. , Issue 98G [page]

1998 Mathematical Reviews
“The Hall number of G, A(G), is the smallest positive integer / such that there is such a system of G-distinct representatives whenever the lists are of length at least /, and Hall’s condition is satisfied  ...  We consider a crude necessary condition for such a system that we call Hall’s condition because of a strong family resemblance to the condition for the existence of an SDR in the famous theorem of P.  ...

### Choice number of complete multipartite graphs K3∗3,2∗(k−5),1∗2 and K4,3∗2,2∗(k−6),1∗3

Wenjie He, Lingmin Zhang, Daniel W. Cranston, Yufa Shen, Guoping Zheng
2008 Discrete Mathematics
Since each k-chromatic graph is a subgraph of a complete k-partite graph, we see that Ohba's conjecture is true if and only if it is true for every complete multipartite graph.  ...  Acknowledgements We would like to express our gratitude to the referees for their careful reading and helpful comments.  ...  Note that G\V satisfies Hall's condition for L ′ . Hence G\V is L ′ -colorable. By hypothesis, X is L| X -colorable.  ...
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