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Some results in square-free and strong square-free edge-colorings of graphs

K.S. Sudeep, Sundar Vishwanathan
<span title="">2007</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Extending the above concept to graphs, a coloring of the edge set E in a graph G(V , E) is called square-free if the sequence of colors on any path in G is square-free.  ...  Algor. 21 (2002) 336-346] who proved bounds on the minimum number of colors needed for a square-free edge-coloring of G on the class of graphs with bounded maximum degree and trees.  ...  Extending the above concepts to graphs, we call a coloring of the edge set E in a graph G(V , E) square-free if the sequence of colors on any path in G is square-free.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disc.2006.09.019">doi:10.1016/j.disc.2006.09.019</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cxn6zvmxabgejbvgaadtcxwlgi">fatcat:cxn6zvmxabgejbvgaadtcxwlgi</a> </span>
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Signified chromatic number of grids is at most 9 [article]

Janusz Dybizbanski
<span title="2019-09-01">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
A signified graph is a pair (G, Σ) where G is a graph, and Σ is a set of edges marked with '-'. Other edges are marked with '+'.  ...  A signified coloring of the signified graph (G, Σ) is a homomorphism into a signified graph (H, Δ). The signified chromatic number of the signified graph (G, Σ) is the minimum order of H.  ...  The signified Paley graph SP q is signified graph (K q , Σ), where K q is the complete graph on vertices F q and set of negative edges Σ = {x, y} : y − x is not a square in F q .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1909.00371v1">arXiv:1909.00371v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dhtx56siuraodgcjrgrtxwx6pi">fatcat:dhtx56siuraodgcjrgrtxwx6pi</a> </span>
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Combinatorial Properties of Triangle-Free Rectangle Arrangements and the Squarability Problem [chapter]

Jonathan Klawitter, Martin Nöllenburg, Torsten Ueckerdt
<span title="">2015</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Using this, we give a new proof that every triangle-free planar graph is the contact graph of such an arrangement.  ...  ., arrangements of interior-disjoint rectangles, with a triangle-free intersection graph.  ...  Another natural question is whether every triangle-free planar graph is a square contact graph.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-27261-0_20">doi:10.1007/978-3-319-27261-0_20</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pwhcmhorijeczenbuv3mmks5vm">fatcat:pwhcmhorijeczenbuv3mmks5vm</a> </span>
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A derivation of an affine plane of order 4 from a triangle-free 3-colored K16

Charles F. Laywine
<span title="">2001</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 variety of results connecting latin squares and graphs of di erent types, are known.  ...  In the derivation each line of this a ne plane is deÿned by a set of 4 vertices of the K16, which are mutually connected by edges of three colors so that each color deÿnes one of three 1-factor of that  ...  [5] showed that with latin squares as vertices and orthogonality between squares as edges, any ÿnite graph is isomorphic to a graph derived from a set of orthogonal latin squares.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(00)00270-3">doi:10.1016/s0012-365x(00)00270-3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5j5hma2tznf4ri65eh3xhq7f2a">fatcat:5j5hma2tznf4ri65eh3xhq7f2a</a> </span>
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2-Edge-Colored Chromatic Number of Grids is at Most 9

Janusz Dybizbański
<span title="2020-03-11">2020</span> <i title="Springer Science and Business Media LLC"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/yxooi3wjmbgqtbp7l4evjeuj4i" style="color: black;">Graphs and Combinatorics</a> </i> &nbsp;
The 2-edge-colored chromatic number of the 2-edge-colored graph ðG; rÞ is the minimum order of H.  ...  A 2-edge-colored coloring of the 2-edge-colored graph ðG; rÞ is a homomorphism into a 2-edgecolored graph ðH; dÞ.  ...  The 2-edge-colored Paley graph SP q is 2-edge-colored graph ðK q ; rÞ, where K q is the complete graph on vertices F q and rðfx; ygÞ ¼ 0 þ 0 () y À x is a non-zero square in F q .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00373-020-02155-y">doi:10.1007/s00373-020-02155-y</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6e2aqr5qxvennhgj64zwkcqszi">fatcat:6e2aqr5qxvennhgj64zwkcqszi</a> </span>
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Nonrepetitively 3-colorable subdivisions of graphs with a logarithmic number of subdivisions per edge [article]

Matthieu Rosenfeld
<span title="2021-02-01">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show that for every graph G and every graph H obtained by subdividing each edge of G at least O(log |V(G)|), H is nonrepetitively 3-colorable.  ...  This answers a question of Wood and improves a similar result of Pezarski and Zmarz that stated the existence of at least one 3-colorable division with a linear number of subdivision vertices per edge.  ...  Given a directed graph D, two sets of colors C and Σ, a nonrepetitive edge coloring φ : A(D) → C of D and a square-free morphism h : C → Σ.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2102.00750v1">arXiv:2102.00750v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/yha2vsr5t5bfpnijxoqrp4zzze">fatcat:yha2vsr5t5bfpnijxoqrp4zzze</a> </span>
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Nonrepetitively 3-Colorable Subdivisions of Graphs with a Logarithmic Number of Subdivisions per edge

Matthieu Rosenfeld
<span title="2021-11-05">2021</span> <i title="The Electronic Journal of Combinatorics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/v5dyak6ulffqfara7hmuchh24a" style="color: black;">Electronic Journal of Combinatorics</a> </i> &nbsp;
We show that for every graph $G$ and every graph $H$ obtained by subdividing each edge of $G$ at least $\Omega(\log |V(G)|)$ times, $H$ is nonrepetitively 3-colorable.  ...  This answers a question of Wood and improves a similar result of Pezarski and Zmarz that stated the existence of at least one 3-colorable subdivision with a linear number of subdivision vertices per edge  ...  Given a directed graph D, two sets of colors C and Σ, a nonrepetitive edge coloring φ : A(D) → C of D and a square-free morphism h : C → Σ.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/10370">doi:10.37236/10370</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/i245z37vznasvhqizp2iudlkea">fatcat:i245z37vznasvhqizp2iudlkea</a> </span>
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Exact square coloring of subcubic planar graphs [article]

Florent Foucaud, Hervé Hocquard, Suchismita Mishra, Narayanan Narayanan, Reza Naserasr, Éric Sopena, Petru Valicov
<span title="2021-01-24">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We prove tight bounds on special subclasses of planar graphs: subcubic bipartite planar graphs and subcubic K 4-minor-free graphs have exact square chromatic number at most 4.  ...  This notion is related to other types of distance-based colorings, as well as to injective coloring. Indeed, for triangle-free graphs, exact square coloring and injective coloring coincide.  ...  Acknowledgements We thank František Kardoš for pointing us out the class of (6, 0)-nanotubes and for insights on fullerene graphs.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2009.00843v3">arXiv:2009.00843v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/74axopvnp5dqxpzdi3behsduda">fatcat:74axopvnp5dqxpzdi3behsduda</a> </span>
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On square-free vertex colorings of graphs

János Barát, Péter Varjú
<span title="">2007</span> <i title="Akademiai Kiado Zrt."> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/w7ssti7ckjhlbh2fvb2enzp4f4" style="color: black;">Studia scientiarum mathematicarum Hungarica (Print)</a> </i> &nbsp;
Following the paper of Alon, Grytczuk, Ha luszczak and Riordan, we examine graph colorings for which the color sequence is square-free on any path.  ...  A century ago Thue showed that there exist arbitrarily long square-free sequences using only three symbols. Sequences can be thought of as colors on the vertices of a path.  ...  Let G be a graph. A vertex coloring of G is called squarefree if the sequence of colors is not a square on any path.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1556/sscmath.2007.1029">doi:10.1556/sscmath.2007.1029</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vxoy5wwyjngx5brt5judnmqr4m">fatcat:vxoy5wwyjngx5brt5judnmqr4m</a> </span>
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Edge Bounds and Degeneracy of Triangle-Free Penny Graphs and Squaregraphs

David Eppstein
<span title="">2018</span> <i title="Journal of Graph Algorithms and Applications"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/4joumle7hffilk276fkubvhxcq" style="color: black;">Journal of Graph Algorithms and Applications</a> </i> &nbsp;
We show that triangle-free penny graphs have degeneracy at most two, list coloring number (choosability) at most three, diameter $D=\Omega(\sqrt n)$, and at most $\min\bigl(2n-\Omega(\sqrt n),2n-D-2\bigr  ...  )$ edges.  ...  The bound on the number of degree-two vertices is tight for square grids. Theorem 2. The degeneracy of every triangle-free penny graph is at most two. Proof.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.7155/jgaa.00463">doi:10.7155/jgaa.00463</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2kr3rhq2cjcuflilbz5owptjzu">fatcat:2kr3rhq2cjcuflilbz5owptjzu</a> </span>
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Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count [chapter]

David Eppstein
<span title="">2018</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We show that triangle-free penny graphs have degeneracy at most two, list coloring number (choosability) at most three, diameter D = Ω( √ n), and at most min 2n − Ω( √ n), 2n − D − 2 edges.  ...  The bound on the number of degree-two vertices is tight for square grids. Theorem 2. The degeneracy of every triangle-free penny graph is at most two. Proof.  ...  Edge count We derive a bound on the number of edges of a triangle-free penny graph by using the isoperimetric theorem to show that the outer face of any representation as a penny graph has many vertices  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-73915-1_39">doi:10.1007/978-3-319-73915-1_39</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2scpbb7ckbajlogaxd53ox5zhq">fatcat:2scpbb7ckbajlogaxd53ox5zhq</a> </span>
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Nonrepetitive colorings of trees

B. Brešar, J. Grytczuk, S. Klavžar, S. Niwczyk, I. Peterin
<span title="">2007</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 coloring of the vertices of a graph G is nonrepetitive if no path in G forms a sequence consisting of two identical blocks.  ...  The proof relies on Thue sequences with additional properties involving palindromic words. We also investigate nonrepetitive edge colorings of trees.  ...  Let f be a nonrepetitive coloring of the edges of T with colors 0, 1, 2. Let e = uv be any edge of T and let P e be the corresponding path with 13 edges joining u and v in the subdivided graph H.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disc.2006.06.017">doi:10.1016/j.disc.2006.06.017</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4d5hizdzs5e77ft6ynwrofzbdm">fatcat:4d5hizdzs5e77ft6ynwrofzbdm</a> </span>
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Combinatorial Properties of Triangle-Free Rectangle Arrangements and the Squarability Problem [article]

Jonathan Klawitter, Martin Nöllenburg, Torsten Ueckerdt
<span title="2015-09-02">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Moreover, we prove that every triangle-free planar graph is the contact graph of such an arrangement.  ...  ., arrangements of interior-disjoint rectangles, with a triangle-free intersection graph.  ...  Another natural question is whether every triangle-free planar graph is a square contact graph.  ... 
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On the computational complexity of strong edge coloring

Mohammad Mahdian
<span title="">2002</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
In the strong edge coloring problem, the objective is to color the edges of the given graph with the minimum number of colors so that every color class is an induced matching.  ...  Acknowledgements The main part of this research was done while I was a graduate student in the Computer Science department of the University of Toronto.  ...  The strong edge coloring problem is, in fact, a special case of the problem of coloring the square of the given graph; it is coloring the square of the line-graph of the graph (see [2] ).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0166-218x(01)00237-2">doi:10.1016/s0166-218x(01)00237-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/poja4zzrv5badmsb3yssc5vecq">fatcat:poja4zzrv5badmsb3yssc5vecq</a> </span>
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Avoiding large squares in trees and planar graphs [article]

Daniel Gonçalves, Pascal Ochem, Matthieu Rosenfeld
<span title="2021-06-03">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The Thue number π(G) of a graph G is the minimum number of colors needed to color G without creating a square on a path of G. For a graph class C, π(C) is the supremum of π(G) over the graphs G∈ C.  ...  Following a suggestion of Grytczuk, we consider the generalized parameters π_k(C) such that only squares of period at least k must be avoided. Thus, π(C)=π_1(C).  ...  We define the sequence U i of outerplanar graphs as follows: • every graph U i has exactly one main edge. • U 0 is K 2 . • U i+1 is obtained from a copy of U i with main edge ab and a copy of U i with  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2106.01521v1">arXiv:2106.01521v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tejs36xy6bgdrezbe2grdularu">fatcat:tejs36xy6bgdrezbe2grdularu</a> </span>
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