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Equitable Division of a Path [article]

Neeldhara Misra, Chinmay Sonar, P. R. Vaidyanathan, Rohit Vaish
<span title="2021-01-26">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We study fair resource allocation under a connectedness constraint wherein a set of indivisible items are arranged on a path and only connected subsets of items may be allocated to the agents.  ...  An allocation is deemed fair if it satisfies equitability up to one good (EQ1), which requires that agents' utilities are approximately equal.  ...  RTI4001 of the Department of Atomic Energy, Government of India. Part of this work was done while RV was supported by the Prof. R Narasimhan postdoctoral award.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2101.09794v2">arXiv:2101.09794v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4zipwfiywff5dpzu26qga4s3ue">fatcat:4zipwfiywff5dpzu26qga4s3ue</a> </span>
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Equitable coloring of interval graphs and products of graphs [article]

Bor-Liang Chen, Ko-Wei Lih
<span title="2009-03-08">2009</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We confirm the equitable Δ-coloring conjecture for interval graphs and establish the monotonicity of equitable colorability for them.  ...  We further obtain results on equitable colorability about square (or Cartesian) and cross (or direct) products of graphs.  ...  We call this representation of G a clique path representation of G. Conversely, the existence of a clique path representation implies that the graph is an interval graph.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0903.1396v1">arXiv:0903.1396v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bazjdddgxzhydjlzodezywj3fi">fatcat:bazjdddgxzhydjlzodezywj3fi</a> </span>
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The starlike trees which span a hypercube

Frank Harary, Martin Lewinter
<span title="">1988</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nkrwe4pmozafvnd72yxufztpku" style="color: black;">Computers and Mathematics with Applications</a> </i> &nbsp;
A connected bipartite graph is called equitable if it has the same number of nodes in each of its two colors. A starlike tree with b branches is a subdivision of the star K~. b with b ~> 3.  ...  This extends a result of Nebesky who had made this observation for b = 3.  ...  Lemma 2 Coloring the nodes of Q~ black and white, given a black node x and a white node y of Q,, there exists a hamiltonian x-y path. Proof.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0898-1221(88)90215-5">doi:10.1016/0898-1221(88)90215-5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/z6fibwtzgrb7ln2tg3lprqtlee">fatcat:z6fibwtzgrb7ln2tg3lprqtlee</a> </span>
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Equitable Allocations of Indivisible Chores [article]

Rupert Freeman, Sujoy Sikdar, Rohit Vaish, Lirong Xia
<span title="2020-02-24">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In addition, we observe that the Leximin solution---which is known to satisfy a strong form of approximate equitability in the goods setting---fails to satisfy even EQ1 for chores.  ...  It does, however, satisfy a novel fairness notion that we call equitability up to any duplicated chore.  ...  Existence of a Simple and Equitable Fair Division: A Short Proof. Mathematical Social Sciences, 87:92-93, 2017. (Cited on page 3) 68(1P1):1-17, 1961.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2002.11504v1">arXiv:2002.11504v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/eyzwjfcbbjfvnfmtftpcgbt6pq">fatcat:eyzwjfcbbjfvnfmtftpcgbt6pq</a> </span>
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Total Equitable List Coloring [article]

Hemanshu Kaul, Jeffrey A. Mudrock, Michael J. Pelsmajer
<span title="2018-03-20">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We prove this conjecture for all graphs satisfying Δ(G) ≤ 2 while also studying the related question of the equitable choosability of powers of paths and cycles.  ...  An equitable coloring is a proper coloring of a graph such that the sizes of the color classes differ by at most one.  ...  graphs where each component is a square of a cycle of length divisible by 3 or a square of a path (Theorem 17).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1803.07450v1">arXiv:1803.07450v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wfw72tiyxna63kzxkitllkarxq">fatcat:wfw72tiyxna63kzxkitllkarxq</a> </span>
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Chore division on a graph

Sylvain Bouveret, Katarína Cechlárová, Julien Lesca
<span title="2019-06-22">2019</span> <i title="Springer Science and Business Media LLC"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7rdrzlhoxjb5ho2u76erh6zpni" style="color: black;">Autonomous Agents and Multi-Agent Systems</a> </i> &nbsp;
We show that deciding the existence of a fair allocation is hard even if the underlying graph is a path or a star. We also present some efficiently solvable special cases for these graph topologies.  ...  We consider three common fair division solution concepts, namely proportionality, envy-freeness and equitability, and two individual disutility aggregation functions: additive and maximum based.  ...  Paths Even if the underlying graph is restricted to be a path then all the considered chore division problems are intractable, as we now show.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10458-019-09415-z">doi:10.1007/s10458-019-09415-z</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qgxgvgqakbd6xkebxhh7r6o4na">fatcat:qgxgvgqakbd6xkebxhh7r6o4na</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200309182212/https://basepub.dauphine.fr/bitstream/handle/123456789/19971/chore_division.pdf;jsessionid=D2BF09CF3D8AF73284B9FCD8F0BC5212?sequence=2" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/3b/29/3b2904e660f7fcf272aeeca4309d03c4cb7fd394.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10458-019-09415-z"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Chore division on a graph [article]

Sylvain Bouveret and Katarína Cechlárová and Julien Lesca
<span title="2018-12-05">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show that deciding the existence of a fair allocation is hard even if the underlying graph is a path or a star. We also present some efficiently solvable special cases for these graph topologies.  ...  We consider three common fair division solution concepts, namely proportionality, envy-freeness and equitability, and two individual disutility aggregation functions: additive and maximum based.  ...  Paths Even if the underlying graph is restricted to be a path then all the considered chore division problems are intractable, as we now show.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1812.01856v1">arXiv:1812.01856v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zry3vxzpcng3tjgntiwaog2ybu">fatcat:zry3vxzpcng3tjgntiwaog2ybu</a> </span>
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Vertex Equitable Labeling of Cyclic Snakes and Bistar Graphs

P. Jeyanthi, A. Maheswari
<span title="2013-12-26">2013</span> <i title="Bangladesh Journals Online (JOL)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7lpoa5kl3je4bpzjmtj6mqhdza" style="color: black;">Journal of Scientific Research</a> </i> &nbsp;
In this paper, we establish vertex equitable labeling of square graph of , nn B and splitting graph of , nn B and kC 4 -snake(k≥1) and the generalized kC n -snake is vertex equitable if n 0(mod4),n≥4.  ...  For aA , let v f (a) be the number of vertices v with f(v) = a.  ...  super division of any path and cycle C n with n ≡ 0 or 3 (mod4) are vertex equitable.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3329/jsr.v6i1.15044">doi:10.3329/jsr.v6i1.15044</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gytbqf7b55ddvgdb64ivm7okxu">fatcat:gytbqf7b55ddvgdb64ivm7okxu</a> </span>
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Page 471 of Communications of the Association for Computing Machinery Vol. 4, Issue 10 [page]

<span title="">1961</span> <i title="Association for Computing Machinery"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_association-for-computing-machinery-communications" style="color: black;">Communications of the Association for Computing Machinery </a> </i> &nbsp;
The Equitable Life Assurance Society of the U.  ...  Titov’s Path Predicted By Philco 2000.  ... 
<span class="external-identifiers"> </span>
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Multicultural Art Education for Social Justice - An Equitable Path to Prosperity for Korea -

장용석
<span title="">2013</span> <i title="Korea Elementary Art Education Association"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/h6ja5cugafe65pqhdg2jrwgvye" style="color: black;">Journal of Art Education</a> </i> &nbsp;
Introduction Over the past 60 years, South Korea has experienced radical changes: Japanese colonization, the Korean War and the division of the country into North Korea and South Korea, rapid economic  ...  A. Park, 2002 J. A. Park, , 2008 . In addition, traditional art is also an important part of the Korean art curriculum.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.35657/jae.2013.34.0.005">doi:10.35657/jae.2013.34.0.005</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/z72w7uvoirewtflrnpd7az2d6a">fatcat:z72w7uvoirewtflrnpd7az2d6a</a> </span>
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A fast algorithm for equitable coloring

Henry A. Kierstead, Alexandr V. Kostochka, Marcelo Mydlarz, Endre Szemerédi
<span title="">2010</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ujm6quky2va7pmzbitmgsrksbu" style="color: black;">Combinatorica</a> </i> &nbsp;
A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most one.  ...  We show that this coloring can be obtained in O(rn 2 ) time, where n is the number of vertices.  ...  Switch witnesses along a W, V − -path to obtain an equitable coloring of G 1 := G[A−W +w ], where w witnesses the first edge of this path.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00493-010-2483-5">doi:10.1007/s00493-010-2483-5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jkosxo4pkjgapht65xurrnic7u">fatcat:jkosxo4pkjgapht65xurrnic7u</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170329060317/http://www.math.uiuc.edu:80/~kostochk/docs/2012/comb10kms.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b6/0d/b60d6b21548ee35ec2909aef0b2dec4bd93defe5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00493-010-2483-5"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

A Short Proof of the Hajnal–Szemerédi Theorem on Equitable Colouring

H. A. KIERSTEAD, A. V. KOSTOCHKA
<span title="2007-07-05">2007</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/domxx2wewzae3e2m2ziqaqbyhm" style="color: black;">Combinatorics, probability &amp; computing</a> </i> &nbsp;
Then |G | is divisible by r + 1 and ∆(G ) ≤ r. Moreover, the restriction of any equitable (r + 1)-coloring of G to G is an equitable (r + 1)coloring of G. Argue by induction on G .  ...  Theorem 4 If G is a graph satisfying ∆(G) ≤ r then G has an equitable (r + 1)-coloring. Proof. We may assume that |G| is divisible by r + 1.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/s0963548307008619">doi:10.1017/s0963548307008619</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/z6nrijvll5g2pnnjzamiaa67ki">fatcat:z6nrijvll5g2pnnjzamiaa67ki</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170706015548/http://cgm.cs.mcgill.ca/~breed/SRL/HS.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/3c/71/3c7198d0d19a453995d11a8aa6a469904400f1c8.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/s0963548307008619"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> cambridge.org </button> </a>

Which Is the Fairest (Rent Division) of Them All?

Ya'akov (Kobi) Gal, Moshe Mash, Ariel D. Procaccia, Yair Zick
<span title="">2016</span> <i title="ACM Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7ljisdx4yfhzbmhados4oxci6y" style="color: black;">Proceedings of the 2016 ACM Conference on Economics and Computation - EC &#39;16</a> </i> &nbsp;
This is not just a theoretical question: many people have used the Spliddit website to obtain envy-free solutions to rent division instances.  ...  We develop a general algorithmic framework that enables the computation of such solutions in polynomial time.  ...  ACKNOWLEDGMENTS This work was supported by EU FP7 FET project, grant agreement n.600854; by the National Science Foundation under grants IIS-1350598, CCF-1215883, and CCF-1525932; and by a Sloan Research  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/2940716.2940724">doi:10.1145/2940716.2940724</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/sigecom/GalMPZ16.html">dblp:conf/sigecom/GalMPZ16</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jbdxwir775g3vki22gltn3m2py">fatcat:jbdxwir775g3vki22gltn3m2py</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20171116184731/http://www.comp.nus.edu.sg:80/~zick/papers/rent.ec16.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/6f/1d/6f1d7be76553270b33c2af6627ccbc435d1450cf.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/2940716.2940724"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> acm.org </button> </a>

Equitable Colorings of Corona Multiproducts of Graphs [article]

Hanna Furmañczyk, Marek Kubale, Vahan V. Mkrtchyan
<span title="2012-10-24">2012</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In particular, we obtain some results regarding the equitable chromatic number for l-corona product G ∘ ^l H, where G is an equitably 3- or 4-colorable graph and H is an r-partite graph, a path, a cycle  ...  A graph is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one.  ...  In our case, when n is not divisible by three, a strongly equitable coloring does not exist.  ... 
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Every tree is 3-equitable

David E. Speyer, Zsuzsanna Szaniszló
<span title="">2000</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
A labeling of a graph is a function f from the vertex set to some subset of the natural numbers. The image of a vertex is called its label.  ...  Cahit as a generalization for graceful labeling. We prove that every tree is 3-equitable.  ...  (A caterpillar is a tree with the property that the removal of the end vertices results in a path.) Szaniszlo proved that every path is k-equitable for any k [6] .  ... 
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