scholarly journals Restricted Extension of Sparse Partial Edge Colorings of Complete graphs

10.37236/9552 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Carl Johan Casselgren ◽  
Lan Anh Pham
Keyword(s):  
Complete Graph ◽  
Edge Coloring ◽  
Positive Answer ◽  
Complete Graphs ◽  
Edge Colorings ◽  
Proper Edge Coloring

Given a partial edge coloring of a complete graph $K_n$ and lists of allowed colors for the non-colored edges of $K_n$, can we extend the partial edge coloring to a proper edge coloring of $K_n$ using only colors from the lists? We prove that this question has a positive answer in the case when both the partial edge coloring and the color lists satisfy certain sparsity conditions.

scholarly journals On Colorful Edge Triples in Edge-Colored Complete Graphs

2020 ◽  
Vol 36 (6) ◽  
pp. 1623-1637
Author(s):  
Gábor Simonyi
Keyword(s):  
Complete Graph ◽  
Edge Coloring ◽  
Complete Graphs ◽  
Edge Colorings ◽  
Minimum Number ◽  
Ramsey Type

Abstract An edge-coloring of the complete graph $$K_n$$ K n we call F-caring if it leaves no F-subgraph of $$K_n$$ K n monochromatic and at the same time every subset of |V(F)| vertices contains in it at least one completely multicolored version of F. For the first two meaningful cases, when $$F=K_{1,3}$$ F = K 1 , 3 and $$F=P_4$$ F = P 4 we determine for infinitely many n the minimum number of colors needed for an F-caring edge-coloring of $$K_n$$ K n . An explicit family of $$2\lceil \log _2 n\rceil $$ 2 ⌈ log 2 n ⌉ 3-edge-colorings of $$K_n$$ K n so that every quadruple of its vertices contains a totally multicolored $$P_4$$ P 4 in at least one of them is also presented. Investigating related Ramsey-type problems we also show that the Shannon (OR-)capacity of the Grötzsch graph is strictly larger than that of the cycle of length 5.

On the Adjacent Vertex Distinguishing Proper Edge Colorings of Several Classes of Complete 5-Partite Graphs

2013 ◽  
Vol 333-335 ◽  
pp. 1452-1455
Author(s):  
Chun Yan Ma ◽  
Xiang En Chen ◽  
Fang Yang ◽  
Bing Yao
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Adjacent Vertex ◽  
Chromatic Numbers ◽  
Edge Colorings ◽  
Proper Edge Coloring ◽  
Minimum Number

A proper $k$-edge coloring of a graph $G$ is an assignment of $k$ colors, $1,2,\cdots,k$, to edges of $G$. For a proper edge coloring $f$ of $G$ and any vertex $x$ of $G$, we use $S(x)$ denote the set of thecolors assigned to the edges incident to $x$. If for any two adjacent vertices $u$ and $v$ of $G$, we have $S(u)\neq S(v)$,then $f$ is called the adjacent vertex distinguishing proper edge coloring of $G$ (or AVDPEC of $G$ in brief). The minimum number of colors required in an AVDPEC of $G$ is called the adjacent vertex distinguishing proper edge chromatic number of $G$, denoted by $\chi^{'}_{\mathrm{a}}(G)$. In this paper, adjacent vertex distinguishing proper edge chromatic numbers of several classes of complete 5-partite graphs are obtained.

scholarly journals Upper density of monochromatic infinite paths

2019 ◽  
Author(s):  
Jan Corsten ◽  
Louis DeBiasio ◽  
Ander Lamaison ◽  
Richard Lang
Keyword(s):  
Complete Graph ◽  
Edge Coloring ◽  
Classical Case ◽  
Infinite Graphs ◽  
Edge Colorings ◽  
Vertex Set ◽  
Upper Density ◽  
Countably Infinite ◽  
Positive Integers ◽  
Red Path

Ramsey Theory investigates the existence of large monochromatic substructures. Unlike the most classical case of monochromatic complete subgraphs, the maximum guaranteed length of a monochromatic path in a two-edge-colored complete graph is well-understood. Gerencsér and Gyárfás in 1967 showed that any two-edge-coloring of a complete graph Kn contains a monochromatic path with ⌊2n/3⌋+1 vertices. The following two-edge-coloring shows that this is the best possible: partition the vertices of Kn into two sets A and B such that |A|=⌊n/3⌋ and |B|=⌈2n/3⌉, and color the edges between A and B red and edges inside each of the sets blue. The longest red path has 2|A|+1 vertices and the longest blue path has |B| vertices. The main result of this paper concerns the corresponding problem for countably infinite graphs. To measure the size of a monochromatic subgraph, we associate the vertices with positive integers and consider the lower and the upper density of the vertex set of a monochromatic subgraph. The upper density of a subset A of positive integers is the limit superior of |A∩{1,...,}|/n, and the lower density is the limit inferior. The following example shows that there need not exist a monochromatic path with positive upper density such that its vertices form an increasing sequence: an edge joining vertices i and j is colored red if ⌊log2i⌋≠⌊log2j⌋, and blue otherwise. In particular, the coloring yields blue cliques with 1, 2, 4, 8, etc., vertices mutually joined by red edges. Likewise, there are constructions of two-edge-colorings such that the lower density of every monochromatic path is zero. A result of Rado from the 1970's asserts that the vertices of any k-edge-colored countably infinite complete graph can be covered by k monochromatic paths. For a two-edge-colored complete graph on the positive integers, this implies the existence of a monochromatic path with upper density at least 1/2. In 1993, Erdős and Galvin raised the problem of determining the largest c such that every two-edge-coloring of the complete graph on the positive integers contains a monochromatic path with upper density at least c. The authors solve this 25-year-old problem by showing that c=(12+8–√)/17≈0.87226.

scholarly journals List Edge-Colorings of Series-Parallel Graphs

10.37236/1474 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Martin Juvan ◽  
Bojan Mohar ◽  
Robin Thomas
Keyword(s):  
Maximum Degree ◽  
Edge Coloring ◽  
Edge Colorings ◽  
Proper Edge Coloring ◽  
Parallel Graph

It is proved that for every integer $k\ge3$, for every (simple) series-parallel graph $G$ with maximum degree at most $k$, and for every collection $(L(e):e\in E(G))$ of sets, each of size at least $k$, there exists a proper edge-coloring of $G$ such that for every edge $e\in E(G)$, the color of $e$ belongs to $L(e)$.

scholarly journals The Dominator Edge Coloring of Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Minhui Li ◽  
Shumin Zhang ◽  
Caiyun Wang ◽  
Chengfu Ye
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Simple Graph ◽  
Edge Colorings ◽  
Color Class ◽  
Proper Edge Coloring ◽  
Minimum Number ◽  
Relationship Of ◽  
The Relationship

Let G be a simple graph. A dominator edge coloring (DE-coloring) of G is a proper edge coloring in which each edge of G is adjacent to every edge of some color class (possibly its own class). The dominator edge chromatic number (DEC-number) of G is the minimum number of color classes among all dominator edge colorings of G , denoted by χ d ′ G . In this paper, we establish the bounds of the DEC-number of a graph, present the DEC-number of special graphs, and study the relationship of the DEC-number between G and the operations of G .

scholarly journals Generalized Hadamard Matrices and 2-Factorization of Complete Graphs

2020 ◽  
pp. 144-151
Author(s):  
W. V. Nishadi ◽  
A. A. I. Perera
Keyword(s):  
Cyclic Group ◽  
Complete Graph ◽  
Hadamard Matrix ◽  
Edge Coloring ◽  
Hadamard Matrices ◽  
Hamiltonian Cycles ◽  
Complete Graphs ◽  
Spanning Subgraph ◽  
Generalized Hadamard Matrix ◽  
Conjugate Transpose

Graph factorization plays a major role in graph theory and it shares common ideas in important problems such as edge coloring and Hamiltonian cycles. A factor  of a graph  is a spanning subgraph of  which is not totally disconnected. An - factor is an - regular spanning subgraph of  and  is -factorable if there are edge-disjoint -factors  such that . We shall refer as an -factorization of a graph . In this research we consider -factorization of complete graph. A graph with  vertices is called a complete graph if every pair of distinct vertices is joined by an edge and it is denoted by . We look into the possibility of factorizing  with added limitations coming in relation to the rows of generalized Hadamard matrix over a cyclic group. Over a cyclic group  of prime order , a square matrix  of order  all of whose elements are the  root of unity is called a generalized Hadamard matrix if , where  is the conjugate transpose of matrix  and  is the identity matrix of order . In the present work, generalized Hadamard matrices over a cyclic group  have been considered. We prove that the factorization is possible for  in the case of the limitation 1, namely, If an edge  belongs to the factor , then the and  entries of the corresponding generalized Hadamard matrix should be different in the   row. In Particular,  number of rows in the generalized Hadamard matrices is used to form -factorization of complete graphs. We discuss some illustrative examples that might be used for studying the factorization of complete graphs.

NEIGHBOR SUM DISTINGUISHING COLORING OF SOME GRAPHS

2012 ◽  
Vol 04 (04) ◽  
pp. 1250047 ◽  
Author(s):  
AIJUN DONG ◽  
GUANGHUI WANG
Keyword(s):  
Planar Graph ◽  
Edge Coloring ◽  
Proper Edge Coloring

A proper [k]-edge coloring of a graph G is a proper edge coloring of G using colors of the set [k] = {1, 2,…,k}. A neighbor sum distinguishing [k]-edge coloring of G is a proper [k]-edge coloring of G such that for each edge uv ∈ E(G), the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. By ndiΣ(G), we denote the smallest value k in such a coloring of G. In this paper, we obtain that (1) ndiΣ(G) ≤ max {2Δ(G) + 1, 25} if G is a planar graph, (2) ndiΣ(G) ≤ max {2Δ(G), 19} if G is a graph such that mad(G) ≤ 5.

Minimal Decompositions of Complete Graphs into Subgraphs with Embeddability Properties

1969 ◽  
Vol 21 ◽  
pp. 992-1000 ◽  
Author(s):  
L. W. Beineke
Keyword(s):  
Projective Plane ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Graphs ◽  
Minimum Number ◽  
Double Torus

Although the problem of finding the minimum number of planar graphs into which the complete graph can be decomposed remains partially unsolved, the corresponding problem can be solved for certain other surfaces. For three, the torus, the double-torus, and the projective plane, a single proof will be given to provide the solutions. The same questions will also be answered for bicomplete graphs.

scholarly journals On the Ramsey Number $R(4,6)$

10.37236/2102 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Fixed Point ◽  
Automorphism Group ◽  
Lower Bound ◽  
Heuristic Search ◽  
Ramsey Number ◽  
Complete Graphs ◽  
Search Procedure ◽  
Edge Colorings

The lower bound for the classical Ramsey number $R(4,6)$ is improved from 35 to 36. The author has found 37 new edge colorings of $K_{35}$ that have no complete graphs of order 4 in the first color, and no complete graphs of order 6 in the second color. The most symmetric of the colorings has an automorphism group of order 4, with one fixed point, and is presented in detail. The colorings were found using a heuristic search procedure.

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