scholarly journals On Colorings Avoiding a Rainbow Cycle and a Fixed Monochromatic Subgraph

10.37236/303 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Maria Axenovich ◽  
JiHyeok Choi
Keyword(s):  
Complete Graph ◽  
Edge Coloring ◽  
Bipartite Graphs ◽  
Fixed Number ◽  
Monochromatic Copy

Let $H$ and $G$ be two graphs on fixed number of vertices. An edge coloring of a complete graph is called $(H,G)$-good if there is no monochromatic copy of $G$ and no rainbow (totally multicolored) copy of $H$ in this coloring. As shown by Jamison and West, an $(H,G)$-good coloring of an arbitrarily large complete graph exists unless either $G$ is a star or $H$ is a forest. The largest number of colors in an $(H,G)$-good coloring of $K_n$ is denoted $maxR(n, G,H)$. For graphs $H$ which can not be vertex-partitioned into at most two induced forests, $maxR(n, G,H)$ has been determined asymptotically. Determining $maxR(n; G, H)$ is challenging for other graphs $H$, in particular for bipartite graphs or even for cycles. This manuscript treats the case when $H$ is a cycle. The value of $maxR(n, G, C_k)$ is determined for all graphs $G$ whose edges do not induce a star.

Ramsey Problems with Bounded Degree Spread

1993 ◽  
Vol 2 (3) ◽  
pp. 263-269 ◽  
Author(s):  
G. Chen ◽  
R. H. Schelp
Keyword(s):  
Positive Integer ◽  
Complete Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Bounded Degree ◽  
Monochromatic Copy

Let k be a positive integer, k ≥ 2. In this paper we study bipartite graphs G such that, for n sufficiently large, each two-coloring of the edges of the complete graph Kn gives a monochromatic copy of G, with some k of its vertices having the maximum degree of these k vertices minus the minimum degree of these k vertices (in the colored Kn) at most k − 2.

scholarly journals The Ramsey number of books

2019 ◽  
Author(s):  
David Conlon
Keyword(s):  
Complete Graph ◽  
Edge Coloring ◽  
The Other ◽  
Ramsey Number ◽  
Natural Question ◽  
Probabilistic Argument ◽  
Complete Subgraph ◽  
Random Construction ◽  
Lower Order Terms ◽  
Monochromatic Copy

Ramsey's Theorem is among the most well-known results in combinatorics. The theorem states that any two-edge-coloring of a sufficiently large complete graph contains a large monochromatic complete subgraph. Indeed, any two-edge-coloring of a complete graph with N=4t−o(t) vertices contains a monochromatic copy of Kt. On the other hand, a probabilistic argument yields that there exists a two-edge-coloring of a complete graph with N=2t/2+o(t) with no monochromatic copy of Kt. Much attention has been paid to improving these classical bounds but only improvements to lower order terms have been obtained so far. A natural question in this setting is to ask whether every two-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of Kt that can be extended in many ways to a monochromatic copy of Kt+1. Specifically, Erdős, Faudree, Rousseau and Schelp in the 1970's asked whether every two-edge-coloring of KN contains a monochromatic copy of Kt that can be extended in at least (1−ok(1))2−tN ways to a monochromatic copy of Kt+1. A random two-edge-coloring of KN witnesses that this would be best possible. While the intuition coming from random constructions can be misleading, for example, a famous construction by Thomason shows the existence of a two-edge-coloring of a complete graph with fewer monochromatic copies of Kt than a random two-edge-coloring, this paper confirms that the intuition coming from a random construction is correct in this case. In particular, the author answers this question of Erdős et al. in the affirmative. The question can be phrased in the language of Ramsey theory as a problem on determining the Ramsey number of book graphs. A book graph B(k)t is a graph obtained from Kt by adding k new vertices and joining each new vertex to all the vertices of Kt. The main result of the paper asserts that every two-edge-coloring of a complete graph with N=2kt+ok(t) vertices contains a monochromatic copy of B(k)t.

scholarly journals On acyclic edge-coloring of complete bipartite graphs

2017 ◽  
Vol 340 (3) ◽  
pp. 481-493
Author(s):  
Ayineedi Venkateswarlu ◽  
Santanu Sarkar ◽  
Sai Mali Ananthanarayanan
Keyword(s):  
Edge Coloring ◽  
Bipartite Graphs ◽  
Acyclic Edge Coloring ◽  
Complete Bipartite Graphs

Colorful Partitions of Cardinal Numbers

1979 ◽  
Vol 31 (3) ◽  
pp. 524-541
Author(s):  
J. Baumgartner ◽  
P. Erdös ◽  
F. Galvin ◽  
J. Larson
Keyword(s):  
Complete Graph ◽  
Edge Coloring ◽  
Cardinal Numbers ◽  
Symmetrizable Space ◽  
Element Set

Use the two element subsets of κ, denoted by [κ]2, as the edge set for the complete graph on κ points. Write CP(κ, µ, v) if there is an edge coloring R: [κ]2 → µ with µ colors so that for every proper v element set X ⊊ κ, there is a point x ∈ κ ∼ X so that the edges between x and X receive at least the minimum of µ and v colors. Write CP⧣(K, µ, v) if the coloring is oneto- one on the edges between x and elements of X.Peter W. Harley III [5] introduced CP and proved that for κ ≧ ω, CP(κ+, κ, κ) holds to solve a topological problem, since the fact that CP(ℵ1, ℵ0, ℵ0) holds implies the existence of a symmetrizable space on ℵ1 points in which no point is a Gδ.

WEIGHTED-EDGE-COLORING OF k-DEGENERATE GRAPHS AND BIN-PACKING

2011 ◽  
Vol 12 (01n02) ◽  
pp. 109-124
Author(s):  
FLORIAN HUC
Keyword(s):  
Bin Packing ◽  
Edge Coloring ◽  
Weighted Graph ◽  
Packing Problem ◽  
Bipartite Graphs ◽  
Specific Class ◽  
Weighted Graphs ◽  
Outerplanar Graphs ◽  
Underlying Graph ◽  
Multiple Edges

The weighted-edge-coloring problem of an edge-weighted graph whose weights are between 0 and 1, consists in finding a coloring using as few colors as possible and satisfying the following constraints: the sum of weights of edges with the same color and incident to the same vertex must be at most 1. In 1991, Chung and Ross conjectured that if G is bipartite, then [Formula: see text] colors are always sufficient to weighted-edge-color (G,w), where [Formula: see text] is the maximum of the sums of the weights of the edges incident to a vertex. We prove this is true for edge-weighted graphs with multiple edges whose underlying graph is a tree. We further generalise this conjecture to non-bipartite graphs and prove the generalised conjecture for simple edge-weighted outerplanar graphs. Finally, we introduce a list version of this coloring together with the list-bin-packing problem, which allows us to obtain new results concerning the original coloring for a specific class of graphs, namely the k-weight-degenerate weighted graph.

Algorithms of Determining any Perfect Matching Mi of Kv

2011 ◽  
Vol 48-49 ◽  
pp. 170-173
Author(s):  
Zhao Di Xu ◽  
Xiao Yi Li ◽  
Wan Xi Chou
Keyword(s):  
Complete Graph ◽  
Perfect Matching ◽  
Edge Coloring ◽  
Round Robin ◽  
Matrix Algorithm

A definition about edge-matrix is given. Two algorithms for solving perfect matching are obtained. Algorithms A is that perfect matching is determined by using edge coloring of edge-matrix ; Algorithm B is that perfect matching is determined by partitioning edge-matrix into sub matrix and also by solving perfect matching of a complete graph .The procedure of constructing round-robin tournament by using algorithm A and round-robin tournament by using algorithm B.

scholarly journals Biclique edge-choosability in some classes of graphs∗

2017 ◽  
Author(s):  
Gabriel A. G. Sobral ◽  
Marina Groshaus ◽  
André L. P. Guedes
Keyword(s):  
Lower Bound ◽  
Edge Coloring ◽  
Bipartite Graphs ◽  
Polynomial Algorithms ◽  
Odd Cycles ◽  
Chordal Bipartite Graphs

In this paper we study the problem of coloring the edges of a graph for any k-list assignment such that there is no maximal monochromatic biclique, in other words, the k-biclique edge-choosability problem. We prove that the K3free graphs that are not odd cycles are 2-star edge-choosable, chordal bipartite graphs are 2-biclique edge-choosable and we present a lower bound for the biclique choice index of power of cycles and power of paths. We also provide polynomial algorithms to compute a 2-biclique (star) edge-coloring for K3-free and chordal bipartite graphs for any given 2-list assignment to edges.

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