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### On graphs critical with respect to vertex partition numbers

Peter Mihók
<span title="">1981</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
For k 3 0, pk(G) den ot e s the Lick-White vertex partition number of G. A graph G is called (n, k)-critical 'f 't I I is connected and for each edge e of G Pk (G -e) < pk (G) = n.  ...  We describe all (2, k&critical graphs and for n 23, k 2 1 we extend and simplify a result of Bollobas and Harary giving one construction of a family of (n, k&critical graphs of every possible order.  ...  The Lick-White vertex partition rzumber, denoted p,JG), is the minimum number of colours required to colour thz vertices of G so that each colour induces a k-degenerate subgraph of G.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(81)90146-1">doi:10.1016/0012-365x(81)90146-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5m7k2uorqzh6fkmvpgtajbupwi">fatcat:5m7k2uorqzh6fkmvpgtajbupwi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170929193049/http://publisher-connector.core.ac.uk/resourcesync/data/elsevier/pdf/804/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS8wMDEyMzY1eDgxOTAxNDYx.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/40/82/40829c6f787a49030e304768efd37bc27f3f5e05.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(81)90146-1"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

### Partition numbers, connectivity and Hamiltonicity

Mieczysław Borowiecki
<span title="">1987</span> <i title="Institute of Mathematics, Czech Academy of Sciences"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lqk6gx24ffgo5cpnsql7ttwg4u" style="color: black;">Časopis pro pěstování matematiky</a> </i> &nbsp;
The author wishes to thank the referee for his valuable comments and suggestions.  ...  If ZP(G) = 7i, while xp(G -v) < n for any vertex v of G, then G is said to be (n, P)critical with respect to the property P.  ...  The unique graph G which is (n, P)-critical with respect to a k-like property P with \V(G)\ ^ (fc + 1) (n -1) + 1 is the complete graph of order (k+ l)(n -1) + 1. Theorem 2.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.21136/cpm.1987.118305">doi:10.21136/cpm.1987.118305</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/h5ww6nobnbel7huqoi7m7nueda">fatcat:h5ww6nobnbel7huqoi7m7nueda</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200814114327/https://dml.cz/bitstream/handle/10338.dmlcz/118305/CasPestMat_112-1987-2_5.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/49/a0/49a000cbbfadb0931814dea1fd31b2c3b851af69.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.21136/cpm.1987.118305"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

### On game chromatic vertex-critical graphs [article]

Marko Jakovac, Daša Štesl
<span title="2021-05-20">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Since the game chromatic number φ(G-x) can either decrease or increase with respect to φ(G), we distinguish between lower, upper and mixed vertex-criticality.  ...  It is shown that χ_g-game-vertex-critical, χ_ig^A-game-vertex-critical and χ_ig^AB-game-vertex-critical graphs are not necessarily connected.  ...  The coloring game-vertex-critical graphs It is well known that vertex-critical graphs (with respect to the usual chromatic number) are connected [26] .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2105.09674v1">arXiv:2105.09674v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/acgen5t7m5hijhgodmgvxpxjyu">fatcat:acgen5t7m5hijhgodmgvxpxjyu</a> </span>
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### Efficient custom instructions generation for system-level design

Huynh Phung Huynh, Yun Liang, Tulika Mitra
<span title="">2010</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/xieukjvqkjdklkdacnla5pbyx4" style="color: black;">2010 International Conference on Field-Programmable Technology</a> </i> &nbsp;
A key component of our solution is an efficient algorithm inspired by multilevel graph partitioning that can quickly produce high-quality custom instructions for the critical regions and thereby alleviate  ...  Customizable embedded processors, where the processor core can be enhanced with application-specific instructions, can provide high performance similar to custom design circuits with the flexibility of  ...  Given Gi = (Vi, Ei) with partitioning solution Pi, a vertex v ∈ Vi is a boundary vertex of partition Pi [v] if it has at least one adjacent vertex u ∈ Vi such that v and u belong to different partitions  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/fpt.2010.5681456">doi:10.1109/fpt.2010.5681456</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/fpt/HuynhLM10.html">dblp:conf/fpt/HuynhLM10</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lppifl6iwnhurmip5q7twfos64">fatcat:lppifl6iwnhurmip5q7twfos64</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20110401075421/http://www.comp.nus.edu.sg/~tulika/fpt10.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/93/c7/93c757d8396f74900fa41c83b597d984b31c85e4.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/fpt.2010.5681456"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

### Chromatic Excellence in Graphs

Kulrekha Mudartha, R. Sundareswaran, V. Swaminathan
<span title="2011-11-25">2011</span> <i title="Christ University Bangalore"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/dk7ojncrk5cgdcdeqs75k66s4i" style="color: black;">Mapana Journal of Sciences</a> </i> &nbsp;
Excellence in graphs introduced by G.H. Fricke is extended to partitions of the vertex set with respect to a parameter.  ...  A graph G is said to be Chromatic excellent if {v} appears in a chromatic partition of G for every vϵV(G). This paper is devoted to the study of chromatic excellence in graphs.  ...  A graph is excellent with respect to a parameter λ (maximum or minimum), if every vertex of the graph belongs to a λ-set of the graph (a λ-set is a maximum/minimum subset of the vertex set with respect  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.12723/mjs.19.5">doi:10.12723/mjs.19.5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nxhtdfnao5dktl66ptpqih2wrm">fatcat:nxhtdfnao5dktl66ptpqih2wrm</a> </span>
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### Turán's theorem andk-connected graphs

Nicolas Bougard, Gwenaël Joret
<span title="">2008</span> <i title="Wiley"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ukzsjb6a6zhyxnjl6nb2mmjc6m" style="color: black;">Journal of Graph Theory</a> </i> &nbsp;
The minimum size of a k-connected graph with given order and stability number is investigated. If no connectivity is required, the answer is given by Turán's Theorem.  ...  We settle the case of 2-connected graphs, characterize the corresponding extremal graphs, and also extend a result of Brouwer related to Turán's Theorem. c (Year) John Wiley & Sons, Inc.  ...  ACKNOWLEDGMENTS We thank Jean-Paul Doignon and Jean Doyen for their valuable comments on an earlier version of this paper.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/jgt.20289">doi:10.1002/jgt.20289</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xebzk5w2srfblmpvhzgntl4hwi">fatcat:xebzk5w2srfblmpvhzgntl4hwi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170808012229/http://www.ulb.ac.be/di/algo/gjoret/turan.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/72/ac/72ac21cb73312fbfa8eae0960a4adb58161b9490.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/jgt.20289"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> wiley.com </button> </a>

### Upper-critical graphs [article]

Jose Antonio Martin H
<span title="2011-04-04">2011</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Analogously, a graph G is called upper-critical if there is no edge that can be added to G such that G preserves its chromatic number, i.e. {e ∈ E(G̅) | χ(G+e) = χ(G) } = ∅.  ...  It is said that G is a critical graph if every element (vertex or edge) of G is critical.  ...  . , it is possible to count the number P (N, k) of upper-critical graphs (i.e. complete k-partite graphs) given the number (N = |V |) of vertices and the chromatic number (k).  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1011.4124v4">arXiv:1011.4124v4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/eglrcv2cgnhqjmh7rzqrnt7rsu">fatcat:eglrcv2cgnhqjmh7rzqrnt7rsu</a> </span>
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### Stability results for graphs with a critical edge

Alexander Roberts, Alex Scott
<span title="">2018</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/54t3hgai4fhhthc74mj7z7tapu" style="color: black;">European journal of combinatorics (Print)</a> </i> &nbsp;
an H-free graph is to being k-partite.  ...  possible number of edges can be made into the k-partite Turán graph by adding and deleting o(n 2 ) edges.  ...  Introduction For n, k ≥ 1, the k-partite Turán graph T k (n) is the complete k-partite graph on n vertices with vertex classes as equal as possible (or equivalently the k-partite graph with maximum number  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.ejc.2018.07.004">doi:10.1016/j.ejc.2018.07.004</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7niificqgjdorlcgm5jzcmnwvu">fatcat:7niificqgjdorlcgm5jzcmnwvu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170324210519/http://people.maths.ox.ac.uk/scott//Papers/stability.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/9a/dc/9adcf0d5e505cec64b6cba700f78e4f270cafd88.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.ejc.2018.07.004"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

### The TensorFlow Partitioning and Scheduling Problem: It's the Critical Path! [article]

Ruben Mayer and Christian Mayer and Larissa Laich
<span title="2017-11-06">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
However, partitioning can not be viewed in isolation. Each device has to select the next graph vertex to be executed, i.e., perform local scheduling decisions.  ...  Our findings indicate that the best partitioning and scheduling heuristics are those that focus on minimizing the execution time of the critical path in the graph.  ...  A vertex v k that is not on the critical path is assigned to the device dev i with minimal sum of execution times of already assigned vertices plus the execution time of v k on dev i (cf.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1711.01912v1">arXiv:1711.01912v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ke7gccuoiben3fnsv2lowpem44">fatcat:ke7gccuoiben3fnsv2lowpem44</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200928215946/https://arxiv.org/pdf/1711.01912v1.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/e5/63/e5635b9050241eef2c5f7a4f809d184c8080d77e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1711.01912v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

### Notes on dual-critical graphs [article]

Zoltán Király, Sándor Kisfaludi-Bak
<span title="2014-10-07">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We define dual-critical graphs as graphs having an acyclic orientation, where the indegrees are odd except for the unique source.  ...  We give equivalent descriptions for dual-critical graphs in the general case, and further equivalent descriptions in the special cases of planar graphs and 3-regular graphs.  ...  Beginning with the second vertex, every vertex has an odd number of predecessors to which it is connected.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1410.1653v1">arXiv:1410.1653v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/prprmmdg25b3xlgszegfymxjsa">fatcat:prprmmdg25b3xlgszegfymxjsa</a> </span>
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### Domatically cocritical graphs

<span title="">1983</span> <i title="Institute of Mathematics, Czech Academy of Sciences"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lqk6gx24ffgo5cpnsql7ttwg4u" style="color: black;">Časopis pro pěstování matematiky</a> </i> &nbsp;
In this paper we shall present some results on graphs which are cocritical with respect to the domatic number. We consider finite undirected graphs without loops and multiple edges.  ...  As 2is a domatic partition of G', any vertex of V(G) -£>* is adjacent to at least one vertex of D d and to at least one vertex of D d+l9 hence to at least two vertices of D*.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.21136/cpm.1983.118161">doi:10.21136/cpm.1983.118161</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/iidazzh62bbxxci6zcx77tmeve">fatcat:iidazzh62bbxxci6zcx77tmeve</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200813143358/https://dml.cz/bitstream/handle/10338.dmlcz/118161/CasPestMat_108-1983-1_8.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/7f/3b/7f3be2c3d9df80b53acef5f20bee2908ecf4e0b4.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.21136/cpm.1983.118161"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

### An Exact 2.9416^n Algorithm for the Three Domatic Number Problem [article]

Tobias Riege, Jörg Rothe
<span title="2005-06-24">2005</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The three domatic number problem asks whether a given undirected graph can be partitioned into at least three dominating sets, i.e., sets whose closed neighborhood equals the vertex set of the graph.  ...  We also present another deterministic and a randomized algorithm for this problem that both have an even better performance for graphs with small maximum degree.  ...  We thank Dieter Kratsch for pointing us to Lawler's algorithm.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/cs/0506090v1">arXiv:cs/0506090v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ssopzsvkdrhllfxyz7s6syfugu">fatcat:ssopzsvkdrhllfxyz7s6syfugu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20171008212042/https://core.ac.uk/download/pdf/2420912.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/70/f9/70f939e5f5df217b07309123dbb6f39769be961b.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/cs/0506090v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

### Co-complete k-Partite Graphs

<span title="2014-04-18">2014</span> <i title="Foundation of Computer Science"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/b637noqf3vhmhjevdfk3h5pdsu" style="color: black;">International Journal of Computer Applications</a> </i> &nbsp;
A co-complete k-partite graph G = (V 1 , V 2 , ..., V k , E), k ≥ 2 is a graph with smallest number k of disjoint parts in which any pair of vertices in the same part are at distance two.  ...  The number of parts in co-complete k-partite graph G is denoted by k(G). In this paper, we initiate a study of this class in graphs and we obtain a characterization for such graphs.  ...  Let G be a co-complete k-partite graph. A vertex v of G is said to be critical vertex if G − v is co-complete (k − 1)-partite graph.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.5120/15906-4939">doi:10.5120/15906-4939</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gqyr236u4zal7fgmu6oeyeg43i">fatcat:gqyr236u4zal7fgmu6oeyeg43i</a> </span>
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### Graph Partitioning and Sparse Matrix Ordering using Reinforcement Learning and Graph Neural Networks [article]

Alice Gatti, Zhixiong Hu, Tess Smidt, Esmond G. Ng, Pieter Ghysels
<span title="2021-06-28">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Our approach is to recursively partition coarser representations of a given graph.  ...  We present a novel method for graph partitioning, based on reinforcement learning and graph convolutional neural networks.  ...  We show that the graph partitioning and vertex separator codes, when trained on graphs of a certain type and with less than 5,000 nodes, generalize well to graphs with many more nodes, as well as to different  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2104.03546v2">arXiv:2104.03546v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/eqhh6muzoje5tpbpn5sjlfnjkq">fatcat:eqhh6muzoje5tpbpn5sjlfnjkq</a> </span>
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### Counterexamples to Grötzsch–Sachs–Koester's conjecture

Andrey A. Dobrynin, Leonid S. Mel'nikov
<span title="">2006</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Let G be a 4-regular planar graph and suppose that G has a cycle decomposition S (i.e., each edge of G is in exactly on cycle of the decomposition) with every pair of adjacent edges on a face always in  ...  Such a graph G arises as a superposition of simple closed curves in the plane with tangencies disallowed.  ...  The technique described above can be directly applied also to the graph shown in Fig. 2 on the right. The constructed graphs of order 18 are vertex 4-critical but not edge 4-critical.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disc.2005.08.010">doi:10.1016/j.disc.2005.08.010</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rijfxng7cndabjrfqadft2rjbe">fatcat:rijfxng7cndabjrfqadft2rjbe</a> </span>
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