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Note – Edge-Coloring Cliques with Three Colors on All 4-Cliques

1998
*
Combinatorica
*

A

doi:10.1007/pl00009822
fatcat:qqokznklprbxbeb3cg2kdvw6pe
*coloring*of the*edges*of K n is constructed such that every copy of K*4*has at least*three**colors**on*its*edges*. As n → ∞, the number of*colors*used is e O( √ log n ) . ...*Color*the*edge*AB*with*the two dimensional vector c(AB) = (c 0 (AB), c 1 (AB)) Fig. 1 : 1 The 2-*colored*K*4*'s Type 1: Here*one**color*class is the path ABCD, while the other is the path BDAC. ... Reversing the labels*on*the path ABCD now puts us back in Case 1. Type 2: Here*one**color*class is the*4*-cycle ABCD, while the other contains the*edges*AC and BD. ...##
###
Clique coloring B_1-EPG graphs
[article]

2017
*
arXiv
*
pre-print

In this paper we prove that B_1-EPG graphs (

arXiv:1602.06723v2
fatcat:5bv22k4ryzdulpmgrzu6g5pi4i
*edge*intersection graphs of paths*on*a grid, where each path has at most*one*bend) are*4*-*clique**colorable*. ... Moreover, given a B_1-EPG representation of a graph, we provide a linear time algorithm that constructs a*4*-*clique**coloring*of it. ... C contains two of these*three**edges*, and every pair of these*three**edges*is contained in at least*one*path P of P (so, it is not an*edge**clique*). ...##
###
Clique coloringB1-EPG graphs

2017
*
Discrete Mathematics
*

In this paper we prove that B 1 -EPG graphs (

doi:10.1016/j.disc.2017.01.019
fatcat:3axljhb3cvg27hrfr7cwbupswy
*edge*intersection graphs of paths*on*a grid, where each path has at most*one*bend) are*4*-*clique**colorable*. ... Moreover, given a B 1 -EPG representation of a graph, we provide a linear time algorithm that constructs a*4*-*clique**coloring*of it. ... of C contains two of these*three**edges*, and every pair of these*three**edges*is contained in at least*one*path P of P (so, it is not an*edge**clique*). ...##
###
The Erdős-Hajnal conjecture for three colors and multiple forbidden patterns
[article]

2021
*
arXiv
*
pre-print

We consider

arXiv:2005.09269v3
fatcat:hq6jzn6zvjhdhl36k4u5bes4oq
*edge*-*colorings**with**three**colors*. ... Specifically, it claims that for any fixed integer k and any*clique*K*on*k vertices*edge*-*colored**with*two*colors*, there is a positive constant a such that in any complete n-vertex graph*edge*-*colored**with*... In addition, there is an H-avoiding*coloring**with*every*clique**on*more than h 2 (n, H) vertices using*all**three**colors*. ...##
###
On the equivalence covering number of splitgraphs

1995
*
Information Processing Letters
*

An equivalence graph is a disjoint union of

doi:10.1016/0020-0190(95)00037-d
fatcat:sythik4kdfgafops5wb574qy3y
*cliques*. For a graph G let eq(G) be the minimum number of equivalence subgraphs of G needed to cover*all**edges*of G. ... Using a similar method we also show that it is NP-complete to decide whether the equivalence coveting number of a graph is 3, even for graphs*with*maximum degree 6 and*with*maximum*clique*number*4*. an ... For each*edge*in K in that*color*class, add*one*special vertex at that*edge*and let that triangle be a*clique*of the equivalence graph. Next consider an*edge*-*coloring*of G*with**three**colors*. ...##
###
The Graph Tessellation Cover Number: Extremal Bounds, Efficient Algorithms and Hardness
[chapter]

2018
*
Lecture Notes in Computer Science
*

A tessellation of a graph is a partition of its vertices into vertex disjoint

doi:10.1007/978-3-319-77404-6_1
fatcat:ufrymdqywze2jozmfuh6w33ifi
*cliques*. A tessellation cover of a graph is a set of tessellations that covers*all*of its*edges*. ... We establish upper bounds*on*the tessellation cover number given by the minimum between the chromatic index of the graph and the chromatic number of its*clique*graph and we show graph classes for which ... A k-*colorable*(resp. k-*edge*-*colorable*) graph is the*one*which admits a*coloring*(resp. an*edge*-*coloring*)*with*at most k*colors*. ...##
###
The tessellation problem of quantum walks
[article]

2017
*
arXiv
*
pre-print

In this work, we focus

arXiv:1705.09014v1
fatcat:xc2wbsnm2jghrhacxzax3qgn4a
*on*a model called staggered quantum walk, which employs advanced ideas of graph theory and has the advantage of including the most important instances of other discrete-time models ... The evolution operator of the staggered model is obtained from a tessellation cover, which is defined in terms of a set of partitions of the graph into*cliques*. ... The maximal*cliques*are the 3-*cliques*of the spanning wheel W 3n , plus*three*new (n + 1)-*cliques*.*All*maximal*cliques*share the vertex*with*label 3n. ...##
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On_the_coloring_of_graphs_formed_by_cliques_sharing_atmost_one_common_point.pdf
[article]

2019
*
Figshare
*

In this work, we try to prove that the chromatic number of the graph formed by adjoining k

doi:10.6084/m9.figshare.8306441.v1
fatcat:6anx4mr7e5hy3fu3yaiswbdwii
*cliques*of order k, any two of which meet at a single vertex is k ... Thus,*all**cliques*have*one*of the vertices*colored**with*the same*color*. ... [1] showed that the conjecture is true when the partial hypergraph S of H determined by the*edges*of size at least*three*can be ∆ S -*edge*-*colored*and satisfies ∆ S ≤ 3. ...##
###
Colored graphs without colorful cycles
[article]

2015
*
arXiv
*
pre-print

We show that these are precisely the graphs which can be iteratively built up from

arXiv:1509.05621v1
fatcat:iuljrol3kzhmphglouwjv566m4
*three*simple*colored*graphs, having 2,*4*, and 5 vertices, respectively. ... A*colored*graph is a complete graph in which a*color*has been assigned to each*edge*, and a*colorful*cycle is a cycle in which each*edge*has a different*color*. ... Then H has at least*three*inner*edges*, since two*edges*only cross once. Hence*all**three**colors*α −1 , α 0 , α 1 must be assigned to inner*edges*of H, and we have once again violated the claim. ...##
###
Colored graphs without colorful cycles

2007
*
Combinatorica
*

We show that these are precisely the graphs which can be iteratively built up from

doi:10.1007/s00493-007-2224-6
fatcat:zr7v5h32izcqtotbzd6cayz4de
*three*simple*colored*graphs, having 2,*4*, and 5 vertices, respectively. ... A*colored*graph is a complete graph in which a*color*has been assigned to each*edge*, and a*colorful*cycle is a cycle in which each*edge*has a different*color*. ... Then H has at least*three*inner*edges*, since two*edges*only cross once. Hence*all**three**colors*α −1 , α 0 , α 1 must be assigned to inner*edges*of H, and we have once again violated the claim. ...##
###
On a conjecture about uniquely colorable perfect graphs

1997
*
Discrete Mathematics
*

In a graph G

doi:10.1016/s0012-365x(96)00353-6
fatcat:7aqdzh3cmnfytg74arfmn5fmoe
*with*maximum*clique*size 09, a*clique*-pair means two*cliques*of size co whose intersection is 09 -1. ... The subject of this paper is the so-called*clique*-pair conjecture (CPC) which states that if a uniquely*colorable*perfect graph is not a*clique*then it contains a*clique*-pair. ... This is a graph G*with*z(G)~>*4*, and, by Fact 11,*all*of its*color*classes have sizes at most*4*. For g(G)>/5, there exist*three**color*classes of the same size. ...##
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Lower Estimate of Clique Size via Edge Coloring

2021
*
Mathematica Pannonica
*

In an earlier work a non-traditional

doi:10.1556/314.2020.00002
fatcat:ttxvsj5pvjcy3od7a2rxyda3w4
*edge**coloring*scheme was proposed to get upper bounds that are typically better than the*one*provided by the well*coloring*of the nodes. ... In this paper we will show that the same scheme for well*coloring*of the*edges*can be used to find lower bounds for the*clique*number of the given graph. ... The so-called monotonic matrices are in intimate connection*with*codes over the alphabet {1, … , }. The code words*all*have length*three*. ...##
###
On the complexity of bicoloring clique hypergraphs of graphs

2002
*
Journal of Algorithms
*

., whether the vertices of G can be

doi:10.1016/s0196-6774(02)00221-3
fatcat:wv3mfkprdfggrnqrbvhe5g7csm
*colored**with*two*colors*so that no maximal*clique*is monochromatic. ... Our two main results say that deciding the bicolorability of C(G) is NP-hard for perfect graphs (and even for those*with**clique*number 3), but solvable in polynomial time for planar graphs. ... Fig. 3 .Lemma*4*.*4*.Corollary*4*. 5 . 34445 Fig. 3. The construction of a G Het b,c and G Hom a,b,c . I .*All**three*heterogeneous bicolorings and the homogeneous*one*. ...##
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Clique-coloring of K_3,3-minor free graphs
[article]

2019
*
arXiv
*
pre-print

The

arXiv:1801.02186v2
fatcat:eooo2nazmrcs7j72nnamxnbhaa
*clique*-chromatic number of G is the least number of*colors*for which G admits a*clique*-*coloring*. ... A*clique*-*coloring*of a given graph G is a*coloring*of the vertices of G such that no maximal*clique*of size at least two is monocolored. ... The remaining case is that*all*vertices in C belong to some other maximal*cliques*and*all**cliques*are in*one*block in T r . ...##
###
Efficient Dynamic Traitor Tracing

2001
*
SIAM journal on computing (Print)
*

*with*

*one*traitor and the remaining four vertices contain

*three*traitors. (

*4*) If the pirate broadcasts a

*color*given to two vertices: Add that

*edge*and repeat step (2) . ... If we get two subsets

*with*

*one*and

*three*traitors, we are done. Otherwise we have a

*4*-good block. ...

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