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

Dhruv Mubayi
1998 Combinatorica
A 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]

Flavia Bonomo, María Pía Mazzoleni, Maya Stein
2017 arXiv   pre-print
In this paper we prove that B_1-EPG graphs (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

Flavia Bonomo, María Pía Mazzoleni, Maya Stein
2017 Discrete Mathematics
In this paper we prove that B 1 -EPG graphs (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]

Maria Axenovich, Richard Snyder, Lea Weber
2021 arXiv   pre-print
We consider 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

A. Blokhuis, T. Kloks
1995 Information Processing Letters
An equivalence graph is a disjoint union of 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]

Alexandre Abreu, Luís Cunha, Tharso Fernandes, Celina de Figueiredo, Luis Kowada, Franklin Marquezino, Daniel Posner, Renato Portugal
2018 Lecture Notes in Computer Science
A tessellation of a graph is a partition of its vertices into vertex disjoint 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]

A. Abreu, L. Cunha, T. Fernandes, C. de Figueiredo, L. Kowada, F. Marquezino, D. Posner, R. Portugal
2017 arXiv   pre-print
In this work, we focus 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.  ...

### On_the_coloring_of_graphs_formed_by_cliques_sharing_atmost_one_common_point.pdf [article]

Ganitarthi
2019 Figshare
In this work, we try to prove that the chromatic number of the graph formed by adjoining k 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]

Richard N. Ball, Aleš Pultr, Petr Vojtěchovský
2015 arXiv   pre-print
We show that these are precisely the graphs which can be iteratively built up from 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

Richard N. Ball, Aleš Pultr, Petr Vojtěchovský
2007 Combinatorica
We show that these are precisely the graphs which can be iteratively built up from 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

Gábor Bacsó
1997 Discrete Mathematics
In a graph G 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.  ...

### Lower Estimate of Clique Size via Edge Coloring

Balázs Király, Sándor Szabó
2021 Mathematica Pannonica
In an earlier work a non-traditional 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

J KRATOCHVIL, Z TUZA
2002 Journal of Algorithms
., whether the vertices of G can be 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.  ...

### Clique-coloring of K_3,3-minor free graphs [article]

Behnaz Omoomi, Maryam Taleb
2019 arXiv   pre-print
The 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

Omer Berkman, Michal Parnas, Jirí Sgall
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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