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A decomposition of gallai multigraphs

Alexander Halperin, Colton Magnant, Kyle Pula (2014)

Discussiones Mathematicae Graph Theory

An edge-colored cycle is rainbow if its edges are colored with distinct colors. A Gallai (multi)graph is a simple, complete, edge-colored (multi)graph lacking rainbow triangles. As has been previously shown for Gallai graphs, we show that Gallai multigraphs admit a simple iterative construction. We then use this structure to prove Ramsey-type results within Gallai colorings. Moreover, we show that Gallai multigraphs give rise to a surprising and highly structured decomposition into directed trees...

A note on the Size-Ramsey number of long subdivisions of graphs

Jair Donadelli, Penny E. Haxell, Yoshiharu Kohayakawa (2005)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Let T s H be the graph obtained from a given graph H by subdividing each edge s times. Motivated by a problem raised by Igor Pak [Mixing time and long paths in graphs, in Proc. of the 13th annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2002) 321–328], we prove that, for any graph H , there exist graphs G with O ( s ) edges that are Ramsey with respect to T s H .

A note on the Size-Ramsey number of long subdivisions of graphs

Jair Donadelli, Penny E. Haxell, Yoshiharu Kohayakawa (2010)

RAIRO - Theoretical Informatics and Applications

Let TsH be the graph obtained from a given graph H by subdividing each edge s times. Motivated by a problem raised by Igor Pak [Mixing time and long paths in graphs, in Proc. of the 13th annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2002) 321–328], we prove that, for any graph H, there exist graphs G with O(s) edges that are Ramsey with respect to TsH.

A Ramsey-style extension of a theorem of Erdős and Hajnal

Peter Komjáth (2001)

Fundamenta Mathematicae

If n, t are natural numbers, μ is an infinite cardinal, G is an n-chromatic graph of cardinality at most μ, then there is a graph X with X ( G ) ¹ μ , |X| = μ⁺, such that every subgraph of X of cardinality < t is n-colorable.

A ramsey-type theorem for multiple disjoint copies of induced subgraphs

Tomoki Nakamigawa (2014)

Discussiones Mathematicae Graph Theory

Let k and ℓ be positive integers with ℓ ≤ k − 2. It is proved that there exists a positive integer c depending on k and ℓ such that every graph of order (2k−1−ℓ/k)n+c contains n vertex disjoint induced subgraphs, where these subgraphs are isomorphic to each other and they are isomorphic to one of four graphs: (1) a clique of order k, (2) an independent set of order k, (3) the join of a clique of order ℓ and an independent set of order k − ℓ, or (4) the union of an independent set of order ℓ and...

A strongly non-Ramsey uncountable graph

Péter Komjáth (1997)

Fundamenta Mathematicae

It is consistent that there exists a graph X of cardinality 1 such that every graph has an edge coloring with 1 colors in which the induced copies of X (if there are any) are totally multicolored (get all possible colors).

Almost-Rainbow Edge-Colorings of Some Small Subgraphs

Elliot Krop, Irina Krop (2013)

Discussiones Mathematicae Graph Theory

Let f(n, p, q) be the minimum number of colors necessary to color the edges of Kn so that every Kp is at least q-colored. We improve current bounds on these nearly “anti-Ramsey” numbers, first studied by Erdös and Gyárfás. We show that [...] , slightly improving the bound of Axenovich. We make small improvements on bounds of Erdös and Gyárfás by showing [...] and for all even n ≢ 1(mod 3), f(n, 4, 5) ≤ n− 1. For a complete bipartite graph G= Kn,n, we show an n-color construction to color the edges...

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