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Restricted list coloring and Hall's condition

2002
*
Discrete Mathematics
*

If L is a

doi:10.1016/s0012-365x(01)00233-3
fatcat:uwjhdtf24re2vobtfnv3nkkwx4
*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*. ...##
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Completing Partial Proper Colorings using Hall's Condition

2015
*
Electronic Journal of Combinatorics
*

In the context of

doi:10.37236/4387
fatcat:vfxkfrzj3zcuddtkuakgye44um
*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*. ...

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

2007
*
International Journal of Mathematics and Mathematical Sciences
*

Hilton

doi:10.1155/2007/72168
fatcat:vb5jojohqzc6xgdjey4moy7wby
*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. ...##
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Author index

2002
*
Discrete Mathematics
*

Goldwasser,

doi:10.1016/s0012-365x(02)00277-7
fatcat:s5wqhobycrgn3kc2st32twlh4m
*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). ...##
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Choosability of a weighted path and free-choosability of a cycle
[article]

2011
*
arXiv
*
pre-print

In this paper, we give necessary

arXiv:1005.5602v3
fatcat:ozezwzkvt5hlpmqugppjz3vttm
*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 ...##
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On choosability of some complete multipartite graphs and Ohba's conjecture

2008
*
Discrete Mathematics
*

These results have been obtained indirectly from the investigation about complete multipartite graphs by Gravier

doi:10.1016/j.disc.2007.03.059
fatcat:dvtcyuei7vbqdhw7c5ua2o2g3m
*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

2009
*
Applied Mathematics Letters
*

In this work we show that if |V (G)| ≤ 2χ (G)+1

doi:10.1016/j.aml.2009.01.001
fatcat:vo7rxi76evdu7azqgcqgdi3uhq
*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. ...##
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Beyond Ohba's Conjecture: A bound on the choice number of k-chromatic graphs with n vertices

2015
*
European journal of combinatorics (Print)
*

We thank Bruce Reed for pointing out a simplification of the proof,

doi:10.1016/j.ejc.2014.08.032
fatcat:25tlzao52vfaxfxz4nggnmlvlu
*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. ...##
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Beyond Ohba's Conjecture: A bound on the choice number of k-chromatic graphs with n vertices
[article]

2014
*
arXiv
*
pre-print

Let ch(G) denote the choice number of a graph G (also called "

arXiv:1308.6739v3
fatcat:mvkpilwvrvfibohf2etffwl33q
*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. ...##
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List colorings with measurable sets

2008
*
Journal of Graph Theory
*

We provide a simpler proof of a measurable generalization of

doi:10.1002/jgt.20335
fatcat:hxi5nxe6tfglngewheelomo7hy
*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. ...##
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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. ...##
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Choice number of complete multipartite graphs K3∗3,2∗(k−5),1∗2 and K4,3∗2,2∗(k−6),1∗3

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

doi:10.1016/j.disc.2007.10.046
fatcat:7ziwqqrrvvgc5dzbpyfle7jgxi
*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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