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On Rainbow Cycles and Proper Edge Colorings of Generalized Polygons [article]

Matt Noble
<span title="2021-06-09">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
An edge coloring of a simple graph G is said to be proper rainbow-cycle-forbidding (PRCF, for short) if no two incident edges receive the same color and for any cycle in G, at least two edges of that cycle  ...  In recent work, Hoffman, et al. study PRCF edge colorings and find many examples of PRCF-bad graphs having girth less than or equal to 4.  ...  Since no cycle in G is rainbow, any 16-cycle has at least two edges colored the same color under ϕ, and it follows that, for any 16-cycle in G, at least two of the sixteen P 10 's that are subgraphs of  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2106.05324v1">arXiv:2106.05324v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5k6lvk666fffbio5z5kviihb5q">fatcat:5k6lvk666fffbio5z5kviihb5q</a> </span>
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Page 1947 of Mathematical Reviews Vol. , Issue 95d [page]

<span title="">1995</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The center subgraph and peripheral subgraph are the subgraphs induced by @(G) and P(G), respectively.  ...  One result here, that ¥(G) —s(G) may be arbitrarily large, solves a problem posed in the paper by Hilton and Johnson [op. cit.].” 95d:05053 05C15 05C10 Korzhik, Vladimir P. [Korzhik, V. P.]  ... 
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On the rainbow connection of Cartesian products and their subgraphs

Sandi Klavžar, Gašper Mekiš
<span title="">2012</span> <i title="Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Gora"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/g64zvtslcjdqncvrfbmvf422qi" style="color: black;">Discussiones Mathematicae Graph Theory</a> </i> &nbsp;
In particular, it is proved that the so-called Θ-coloring of an isometric subgraph of a hypercube is its unique optimal c-strong rainbow connected coloring.  ...  The concept of c-strong rainbow connected coloring is introduced.  ...  There P 20 is a subgraph of P 5 P 4 , rc(P 20 ) = 19, and rc(P 5 P 4 ) = rc(P 5 ) + rc(P 4 ) = 7 by Proposition 1. Of course, the example easily generalizes to P n P m for any n and m.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.7151/dmgt.1644">doi:10.7151/dmgt.1644</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qtu4d7cuj5fzxp26nk2i4r7xxi">fatcat:qtu4d7cuj5fzxp26nk2i4r7xxi</a> </span>
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Page 922 of Mathematical Reviews Vol. , Issue 2000b [page]

<span title="">2000</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Borodin (RS-AOSSI; Novosibirsk) 2000b:05055 05C15 Erdés, Paul [Erdés, Paul'] (H-AOS; Budapest); Fowler, Tom (1-GAIT; Atlanta, GA) Finding large p-colored diameter two subgraphs.  ...  We show for k >1 and k/2<p<k that there is always a p-colored diameter-two subgraph of K, containing at least COMBINATORICS 922 (k + p)n/2k vertices and that this is best possible up to an addi- tive constant  ... 
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Page 798 of Mathematical Reviews Vol. 58, Issue 2 [page]

<span title="">1979</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Further, we prove that a complete edge-colored graph with n vertices contains a complete subgraph on p vertices in which no two edges have the same color if (§) >(4)=4_, (9), where e; is the number of  ...  Author’s summary: “Let a(k, p, h) be the maximum number of vertices a complete edge-colored graph may have with no color appearing more than k times at any vertex and not containing a complete subgraph  ... 
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Cellular graph automata. II. graph and subgraph isomorphism, graph structure recognition

Angela Wu, Azriel Rosenfeld
<span title="">1979</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/fbkexx2qg5anljkxraobuswdq4" style="color: black;">Information and Control</a> </i> &nbsp;
It discusses acceptance tasks that involve the detection of graph or subgraph isomorphism in time proportional to the diameter of the given graph.  ...  We have been unable to find a diameter time cellular d-graph acceptor that can decide subgraph isomorphism for general d-graphs.  ...  Therefore the coloring process takes diameter(F) time. | k-Locally-Homogeneous d-Graphs As pointed out in Section 3, the difficulty in obtaining diameter-time algorithms for subgraph isomorphism seems  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0019-9958(79)90296-1">doi:10.1016/s0019-9958(79)90296-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4zd4obna4beo5mxa2yoehsw22q">fatcat:4zd4obna4beo5mxa2yoehsw22q</a> </span>
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Graph theory in the information age [chapter]

<span title="2006-08-28">2006</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ocj45idmybftzkw6tvytqr4xiq" style="color: black;">American Mathematical Society. CBMS Regional Conference Series in Mathematics</a> </i> &nbsp;
However, in a recent paper by Liben-Nowell and Kleinberg [52] , it was observed that the tree-like subgraphs derived from some chain-letter data seem to have relatively large diameter.  ...  In the other direction, if p < (1 − ǫ)/d, then almost surely the percolated subgraph G p contains no giant component.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/cbms/107/01">doi:10.1090/cbms/107/01</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pmsgz2juonfdhmqnbvob3fjzby">fatcat:pmsgz2juonfdhmqnbvob3fjzby</a> </span>
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On the multiple Borsuk numbers of sets

Mihály Hujter, Zsolt Lángi
<span title="2013-10-10">2013</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/whoel4vntnhddbjlrtj3w2jgra" style="color: black;">Israel Journal of Mathematics</a> </i> &nbsp;
The Borsuk number of a set S of diameter d >0 in Euclidean n-space is the smallest value of m such that S can be partitioned into m sets of diameters less than d.  ...  Our aim is to generalize this notion in the following way: The k-fold Borsuk number of such a set S is the smallest value of m such that there is a k-fold cover of S with m sets of diameters less than  ...  p b and p r = −p b that are colored differently, and a diameter of S, connecting p and q, corresponds to the two edges p b q r and p r q b .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s11856-013-0048-1">doi:10.1007/s11856-013-0048-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2lce6klv2zedxgva3vovunpys4">fatcat:2lce6klv2zedxgva3vovunpys4</a> </span>
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Decompositions into subgraphs of small diameter [article]

Jacob Fox, Benny Sudakov
<span title="2009-06-18">2009</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We also prove that if G has large minimum degree, all the edges of G can be covered by a small number of low diameter subgraphs.  ...  We investigate decompositions of a graph into a small number of low diameter subgraphs.  ...  If the current graph has at least m/2 edges, then by the above lemma we can find a subgraph of diameter at most 3 with at least (m/2) 3 32n 4 edges.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0906.3530v1">arXiv:0906.3530v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7vkgpjn5kjggdfr4aqu2torhuy">fatcat:7vkgpjn5kjggdfr4aqu2torhuy</a> </span>
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Induced subtrees in graphs of large chromatic number

A. Gyárfás, E. Szemeredi, Zs. Tuza
<span title="">1980</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
The main ;csult concerns the case when T has radius two.  ...  paper proves special eases of the following conjecture: for any fined tr~,e "J~ there exists a natural number f = fiT) ~o that every triangle-free graph of chromatic number ] T) contains T as au induced subgraph  ...  If G is a triangle-free graph of diameter two, x(G)>~R(k, 3)+ 1 and Pe V(G), then G contains Tk,~ as an induced subgraph, so that P is the center qf Pre~f.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(80)90230-7">doi:10.1016/0012-365x(80)90230-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5ijn3rekffhjhn4xrv4sqyvdq4">fatcat:5ijn3rekffhjhn4xrv4sqyvdq4</a> </span>
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Solving the Maximum Clique and Vertex Coloring Problems on Very Large Sparse Networks

Anurag Verma, Austin Buchanan, Sergiy Butenko
<span title="">2015</span> <i title="Institute for Operations Research and the Management Sciences (INFORMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ko3yoz32wja7tmqwbgqqjmfyxa" style="color: black;">INFORMS journal on computing</a> </i> &nbsp;
This paper explores techniques for solving the maximum clique and vertex coloring problems on very large-scale real life networks.  ...  Solving the maximum clique and vertex coloring problems on very large sparse networks. INFORMS Journal on Computing, 27(1):164-177, 2015. http://dx.doi.org/10.  ...  Acknowledgements We would like to thank an Associate Editor and two reviewers for their suggestions that helped to significantly improve the paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1287/ijoc.2014.0618">doi:10.1287/ijoc.2014.0618</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/db3r6flnebbbnh6uo2lgsrk6pu">fatcat:db3r6flnebbbnh6uo2lgsrk6pu</a> </span>
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On coloring the arcs of a tournament, covering shortest paths, and reducing the diameter of a graph

Nili Guttmann-Beck, Refael Hassin
<span title="">2011</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/s6sw62lf6zcxrghfhvjqhycu5y" style="color: black;">Discrete Optimization</a> </i> &nbsp;
We define closed edge colorings of directed graphs, and state a conjecture about the maximum size of a tournament graph that can be arc-colored with m colors and contain no closed subgraphs.  ...  We use the latter property to produce an approximation algorithm with improved bound for minimizing the diameter or the radius of an unweighted graph by adding to it a given number of new edges.  ...  If G has p+2 nodes with distance at least R between any two of them, then Conjecture 2.2 implies that using p edges the optimal solution for the p-edge minimum diameter problem is at least R and the optimal  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disopt.2010.09.008">doi:10.1016/j.disopt.2010.09.008</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4v2uy3qj6zafvamfoqtejs5oui">fatcat:4v2uy3qj6zafvamfoqtejs5oui</a> </span>
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The Balanced Connected Subgraph Problem [article]

Sujoy Bhore, Sourav Chakraborty, Satyabrata Jana, Joseph S. B. Mitchell, Supantha Pandit, Sasanka Roy
<span title="2018-09-24">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
., planar graphs, chordal graphs, trees, split graphs, bipartite graphs with a proper red/blue 2-coloring, and graphs with diameter 2.  ...  In this paper, we consider the following Balanced Connected Subgraph (shortly, BCS) problem. The input is a graph G=(V,E), with each vertex in the set V having an assigned color, "red" or "blue".  ...  Then, from G ′ , we will find the desired balanced subgraph with |S B ∪ K B | many vertices of each color. Moreover, this process is done in two steps.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1809.08856v1">arXiv:1809.08856v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bi33ap2yzjcoljtc7xla7oapmy">fatcat:bi33ap2yzjcoljtc7xla7oapmy</a> </span>
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Finding Large Monochromatic Diameter Two Subgraphs [article]

Tom Fowler
<span title="1999-08-30">1999</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Given a coloring of the edges of the complete graph on n vertices in k colors, by considering the neighbors of an arbitrary vertex it follows that there is a monochromatic diameter two subgraph on at least  ...  We show that for k > 3 this is asymptotically best possible, and that for k=2 there is always a monochromatic diameter two subgraph on at least 3 4 n vertices, which again, is best possible.  ...  Now we show that any edge two-coloring of K n will render a monochromatic diameter two subgraph of order at least ⌈ 3n 4 ⌉.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/9908170v1">arXiv:math/9908170v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/m625cwqpz5bqrjo6fhrshg5bqq">fatcat:m625cwqpz5bqrjo6fhrshg5bqq</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-math9908170/math9908170.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> File Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/9908170v1" 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>

A Ramsey-type result for the hypercube

Noga Alon, Radoš Radoičić, Benny Sudakov, Jan Vondrák
<span title="">2006</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;
Our techniques provide also a characterization of all subgraphs H of the hypercube which are Ramsey, i.e., have the property that for every k, any k-edge coloring of a sufficiently large Q n contains a  ...  We prove that for every fixed k and ≥ 5 and for sufficiently large n, every edge coloring of the hypercube Q n with k colors contains a monochromatic cycle of length 2 .  ...  Note that the number of colors needed in both cases is equal to the diameter of the subgraph in question.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/jgt.20181">doi:10.1002/jgt.20181</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nno37vf3dnf65cbq5plrpeslum">fatcat:nno37vf3dnf65cbq5plrpeslum</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20131123124408/http://www.math.ias.edu/csdm/files/04-05/nalon_a_ramsey_type_result_for_the_hypercube.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/b5/47/b547ceab4b7fac7960346c00d430c4c97366afa9.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/jgt.20181"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> wiley.com </button> </a>
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