(Recasts after Park speech, visit to cemetery)
By Jack Kim and David Chance
SEOUL, Dec 20 (Reuters) - South Korea's President-elect, Park Geun-hye, used her first major speech on Thursday to warn of the risks posed by a hostile North Korea and also fired a political shot across the bows of Japan's incoming Prime Minister Shinzo Abe.
Speaking after a visit to the country's national cemetery, which included a poignant homage at the graves of her assassinated father and mother, South Korea's first female leader pledged to spread wealth more evenly.
Park has said she will hold talks with North Korea and resume aid to the isolated and belligerent country, but only if it abandons its nuclear weapons programme. The impoverished North launched a rocket last week that critics said was a test for technology that could be used for a long-range missile that could one day carry a nuclear warhead.
"North Korea's long-range missile launch showed how grave the security reality is that we are faced with," Park told a news conference a day after her convincing election win.
Park will take office in February and signalled she would continue outgoing Lee Myung-bak's tough line on territorial claims that Japan has on South Korea.
The relationship between them, the two closest allies of the United States in the region, has been damaged by an island row and the issue of an apology and compensation from Japan for the forced sexual slavery of Korean women in World War Two.
South Korea says Japan, which has similar disputes with China, has not come to terms with its harsh past rule of Korea. Japan says it has paid compensation for the slavery issue and has apologised.
"I will try to work for greater reconciliation, cooperation and peace in North East Asia based on correct perception of history," she said in an apparent reference to the simmering conflict with Tokyo.
Park, 60, replaces fellow conservative Lee after his mandatory single, five-year term ends.
The slightly built and elegant Park grew up in Seoul's presidential palace during the 18-year rule of her father, Park Chung-hee, who took power in a military coup in 1961.
Park on Tuesday called for national "reconciliation" in South Korea and pledged again to share wealth more evenly, but offered no clues about how she would implement policies.
She is likely to face protests by South Korea's vocal left, angry over the rise to power of the daughter of a man they believe was a repressive "dictator".
"This will be a tremendous burden on her ability to govern," political commentator Yu Chang-seon said of Park's heritage.
"It effectively means that she could be in direct conflict with half of society ... The first six months will be key."
On the economy, which dominated the election campaign, Park has promised more social welfare but given few specifics.
Korea has achieved astonishing success in rising from the ashes of the 1950-53 Korean War to become the world's 14th largest economy, but rewards have been thinly spread.
Economic growth was 5.5 percent for decades, driven by some of the world's biggest companies, such as Samsung Electronics Co Ltd and Hyundai Motor Co. That pace has slowed and this year the economy will expand by about 2 percent.
The hundreds of thousands of graduates churned out by South Korean universities each year complain they have trouble finding decent jobs and income differentials have widened sharply.
Park has at times invoked her father's legacy of rapid growth that propelled South Korea into the league of industrialised nations.
At other times, she has apologised for his suppression of protests and the execution of people suspected of sympathising with the North, which is still technically at war with the South after an armistice ended the Korean War.
Families of those who were executed under her father's rule believe Park has not apologised enough and that she has sought to sweep her past under the carpet. Park was her father's "First Lady" following the 1974 assassination of her mother up until her father was also shot and killed in 1979.
The most notorious executions under Park Chung-hee's rule were of eight men dubbed the "People's Revolutionary Party". They were hanged 24 hours after being sentenced for treason.
The eight, aged 30 to 52, represented a broad section of South Korean society, comprising a bee keeper, a brewery owner, an acupuncturist and teachers. They were exonerated posthumously by the Supreme Court in 2007.
"What she needs to be doing is to reach out to everyone, to those who oppose her, to show her interest and offer her sympathy and to say that she feels sorry for what happened," said Reverend Park Jung-il, who was chief army chaplain in April 1975 and witnessed the dawn executions of the eight men.
As well as confronting a domestic legacy that is still painful for many South Koreans, Park will have to deal with Kim Jong-un, the 29-year-old ruler of North Korea whose grandfather ordered several assassination attempts on her father.
During a 2002 thaw in relations, Park met Kim Jong-il, the father of the latest Kim to rule the North, which in 2010 sank a South Korean naval vessel and shelled a South Korean island.
Park has said she will seek to improve ties with Pyongyang.
Lee, the outgoing president, infuriated the North by cutting off aid to a country where a third of the population is said by the United Nations to be malnourished.
On the face of it, North Korea is in no mood for compromise. It has declared it will not ditch its nuclear weapons capacity, which it recently termed "treasured".
It pushed ahead with last week's rocket launch, despite it being banned under U.N. resolutions imposed in the wake of its 2006 and 2009 nuclear tests, as the South got ready to vote.
Park herself has become a target for Pyongyang's propaganda machine, which has denounced Lee's five-year rule for bringing "nightmare, despair, (and )catastrophe". (Editing by Dean Yates and Paul Tait)