By default on CentOS 6.x is used 8169 instead of 8168 driver for Ethernet.

Network device display (lspci output):

[root@static89 ~]# lspci|grep Eth
03:00.0 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller (rev 09)

Loaded driver:

[root@static89 ~]# lsmod|grep 816
r8169                  74378  0

From dmesg output:

r8169 Gigabit Ethernet driver 2.3LK-NAPI loaded
r8169 0000:03:00.0: PCI INT A -> GSI 16 (level, low) -> IRQ 16
r8169 0000:03:00.0: setting latency timer to 64
r8169 0000:03:00.0: irq 34 for MSI/MSI-X

For installing rtl8168 driver:

rpm --import https://www.elrepo.org/RPM-GPG-KEY-elrepo.org
rpm -Uvh http://www.elrepo.org/elrepo-release-6-8.el6.elrepo.noarch.rpm
yum update
yum install kmod-r8168

After reboot:

[root@static89 ~]# dmesg |grep 8168
r8168 Gigabit Ethernet driver 8.040.00-NAPI loaded
r8168 0000:03:00.0: PCI INT A -> GSI 16 (level, low) -> IRQ 16
r8168 0000:03:00.0: setting latency timer to 64
r8168 0000:03:00.0: irq 34 for MSI/MSI-X
r8168: This product is covered by one or more of the following patents: US6,570,884, US6,115,776, and US6,327,625.
r8168  Copyright (C) 2015  Realtek NIC software team 
r8168: eth0: link up

For Centos 7 (updated, 11.04.2017):

rpm --import https://www.elrepo.org/RPM-GPG-KEY-elrepo.org
rpm -Uvh http://www.elrepo.org/elrepo-release-7.0-2.el7.elrepo.noarch.rpm
yum update
yum install kmod-r8168
init 6 # reboot