# Triangle-free partial graphs and edge covering theorems

J. Lehel, Z. Tuza

Research output: Contribution to journalArticle

18 Citations (Scopus)

### Abstract

In section 1 some lower bounds are given for the maximal number of edges ofa (p - 1)- colorable partial graph. Among others we show that a graph on n vertices with m edges has a (p-1)-colorable partial graph with at least mTn.p/(n2) edges, where Tn.p denotes the so called Turán number. These results are used to obtain upper bounds for special edge covering numbers of graphs. In Section 2 we prove the following theorem: If G is a simple graph and μ is the maximal cardinality of a triangle-free edge set of G, then the edges of G can be covered by μ triangles and edges. In Section 3 related questions are examined.

Original language English 59-65 7 Discrete Mathematics 39 1 https://doi.org/10.1016/0012-365X(82)90040-1 Published - 1982

### Fingerprint

Triangle-free
Covering
Partial
Graph in graph theory
Theorem
Covering number
Simple Graph
Triangle
Cardinality
Lower bound
Upper bound
Denote

### ASJC Scopus subject areas

• Discrete Mathematics and Combinatorics
• Theoretical Computer Science

### Cite this

In: Discrete Mathematics, Vol. 39, No. 1, 1982, p. 59-65.

Research output: Contribution to journalArticle

@article{6ba9745165d34180922cfcb128ca7876,
title = "Triangle-free partial graphs and edge covering theorems",
abstract = "In section 1 some lower bounds are given for the maximal number of edges ofa (p - 1)- colorable partial graph. Among others we show that a graph on n vertices with m edges has a (p-1)-colorable partial graph with at least mTn.p/(n2) edges, where Tn.p denotes the so called Tur{\'a}n number. These results are used to obtain upper bounds for special edge covering numbers of graphs. In Section 2 we prove the following theorem: If G is a simple graph and μ is the maximal cardinality of a triangle-free edge set of G, then the edges of G can be covered by μ triangles and edges. In Section 3 related questions are examined.",
author = "J. Lehel and Z. Tuza",
year = "1982",
doi = "10.1016/0012-365X(82)90040-1",
language = "English",
volume = "39",
pages = "59--65",
journal = "Discrete Mathematics",
issn = "0012-365X",
publisher = "Elsevier",
number = "1",

}

TY - JOUR

T1 - Triangle-free partial graphs and edge covering theorems

AU - Lehel, J.

AU - Tuza, Z.

PY - 1982

Y1 - 1982

N2 - In section 1 some lower bounds are given for the maximal number of edges ofa (p - 1)- colorable partial graph. Among others we show that a graph on n vertices with m edges has a (p-1)-colorable partial graph with at least mTn.p/(n2) edges, where Tn.p denotes the so called Turán number. These results are used to obtain upper bounds for special edge covering numbers of graphs. In Section 2 we prove the following theorem: If G is a simple graph and μ is the maximal cardinality of a triangle-free edge set of G, then the edges of G can be covered by μ triangles and edges. In Section 3 related questions are examined.

AB - In section 1 some lower bounds are given for the maximal number of edges ofa (p - 1)- colorable partial graph. Among others we show that a graph on n vertices with m edges has a (p-1)-colorable partial graph with at least mTn.p/(n2) edges, where Tn.p denotes the so called Turán number. These results are used to obtain upper bounds for special edge covering numbers of graphs. In Section 2 we prove the following theorem: If G is a simple graph and μ is the maximal cardinality of a triangle-free edge set of G, then the edges of G can be covered by μ triangles and edges. In Section 3 related questions are examined.

UR - http://www.scopus.com/inward/record.url?scp=0040321493&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040321493&partnerID=8YFLogxK

U2 - 10.1016/0012-365X(82)90040-1

DO - 10.1016/0012-365X(82)90040-1

M3 - Article

AN - SCOPUS:0040321493

VL - 39

SP - 59

EP - 65

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1

ER -