Abstract
AbstractLet $$\mathrm{pr}(K_{n}, G)$$
pr
(
K
n
,
G
)
be the maximum number of colors in an edge-coloring of $$K_{n}$$
K
n
with no properly colored copy of G. For a family $${\mathcal {F}}$$
F
of graphs, let $$\mathrm{ex}(n, {\mathcal {F}})$$
ex
(
n
,
F
)
be the maximum number of edges in a graph G on n vertices which does not contain any graphs in $${\mathcal {F}}$$
F
as subgraphs. In this paper, we show that $$\mathrm{pr}(K_{n}, G)-\mathrm{ex}(n, \mathcal {G'})=o(n^{2}), $$
pr
(
K
n
,
G
)
-
ex
(
n
,
G
′
)
=
o
(
n
2
)
,
where $$\mathcal {G'}=\{G-M: M \text { is a matching of }G\}$$
G
′
=
{
G
-
M
:
M
is a matching of
G
}
. Furthermore, we determine the value of $$\mathrm{pr}(K_{n}, P_{l})$$
pr
(
K
n
,
P
l
)
for sufficiently large n and the exact value of $$\mathrm{pr}(K_{n}, G)$$
pr
(
K
n
,
G
)
, where G is $$C_{5}, C_{6}$$
C
5
,
C
6
and $$K_{4}^{-}$$
K
4
-
, respectively. Also, we give an upper bound and a lower bound of $$\mathrm{pr}(K_{n}, K_{2,3})$$
pr
(
K
n
,
K
2
,
3
)
.
Funder
China Sponsorship Council
Nemzeti Kutatási Fejlesztési és Innovációs Hivatal
ELKH Alfréd Rényi Institute of Mathematics
Publisher
Springer Science and Business Media LLC
Subject
Discrete Mathematics and Combinatorics,Theoretical Computer Science
Reference21 articles.
1. Alon, N.: On a conjecture of Erdős, Simonovits and Sós concerning anti-Ramsey theorems. J. Graph Theory 7(1), 91–94 (1983)
2. Axenovich, M., Jiang, T.: Anti-Ramsey numbers for small complete bipartite graphs. Ars Combin. 73, 311–318 (2004)
3. Bialostocki, A., Gilboa, S., Roditty, Y.: Anti-Ramsey number of small graphs. Ars Combin. 123, 41–53 (2015)
4. Brown, W.G.: On graphs that do not contain a Thomsen graph. Can. Math. Bull. 9, 281–285 (1966)
5. Erdős, P., Gallai, T.: On maximal paths and circuits of graphs. Acta Math. Acad. Sci. Hung. 10, 337–356 (1959)