Which brings us to the 1945 and 1966 seasons, which are certainly years where Alabama could accept championships that were awarded by several systems. Those teams did go undefeated and did win major bowl games.
Alabama's 1945 team, starring Harry Gilmer, went 9-0 in regular season and then trounced USC, 34-14, in the Rose Bowl. There were jokes that Bama beating Southern Cal like that might make the Rose Bowl quit inviting the Tide. Some joke! That's what happened! The Rose Bowl became a closed shop, nothing but Pac-8 (then) vs. Big Ten. That 1945 team finished undefeated and with a Rose Bowl victory, but finished only second in the nation to non-bowling Army, which had stockpiled many of the country's finest players during World War II. The Tide outscored the opposition by 430-80.
The 1966 Alabama team was going for a third consecutive national championship and that Bama squad of Coach Paul Bryant did all it could. What it couldn't do was overcome the hypocrisy of the Associated Press. The AP had decided to award its trophy after the bowl games following the 1964 Alabama team's loss to Texas in the Orange Bowl. Bama came back in the Orange Bowl following the 1965 season and won the title because of the policy change, defeating Nebraska.
But in 1966 Notre Dame and Michigan State played to a 10-10 tie, with the "Fighting" Irish happy to settle for the non-win. AP voters validated Notre Dame Coach Ara Parseghian's abandonment of "Win one for the Gipper" by voting the Irish number one and Michigan State number two. Moreover, the AP decided to revert to its policy of pre-bowl completion of the poll because Notre Dame didn't go to bowls in those days (it was against Notre Dame's academic policy until the bowl payoffs got to be big enough to overcome that virtue) and Michigan State was barred from bowl play that year by a Big Ten rule.
As for Alabama, the Tide went through the regular season with a 10-0 record and then romped over Nebraska by 34-7 with Kenny Stabler's passes to Ray Perkins doing particular damage. That Bama squad outscored opponents by 301-44.
There is at least one other team that was denied the national championship against all reason. In 1977 Alabama opened the season with a 34-13 win over Ole Miss. (That will be important in a moment.) In the second game of the year, the Tide lost by a touchdown at Nebraska, so it was not and undefeated, untied team. Later Bama would defeat No. 1 ranked Southern Cal in Los Angeles, enabling Texas to take over the top spot.
After the bowl games that year, in which Alabama downed Ohio State in the Sugar Bowl and Notre Dame beat Texas in the Cotton Bowl, the Irish poll vaulted, from fifth to first, over Alabama, which finished second. All the contenders had a loss that year. Notre Dame's loss was to that Ole Miss team Alabama had romped.
Incidentally, had Alabama won that 1977 national championship, it would have preceded titles won in 1978 and 1979. No team has ever won three in a row.
I would say that Alabama has made the correct decision to not claim those three championships – 1945, 1966, 1977. I also believe the 1966 and 1977 teams were deserving of the titles that were not awarded. The 1945 team was almost certainly the nation's best pure college football team, but no one can blame Army for having beefed up at West Point for World War II.
As for Meyer, who is now coach at Ohio State, contending that 2008 Florida is the greatest national championship team, consider that the Gators were not undefeated. Florida lost to Ole Miss. In Gainesville. That eliminates the Gators from "great" by the same criterion that scars 1941 Bama. You cannot lose to a team from Mississippi and be great. The primary claim for Meyer is that his team beat No. 1 ranked Alabama in the SEC Championship Game.
When Auburn decided to add three championships – 1913, 1983, and 1993 – the Tigers also had to accept the derision. Most of Alabama's pre-poll championships are based on winning the Rose Bowl, which purportedly matched the best team from the East against the best of the West. Other college football giants also claim such titles.
Auburn's 1913 team did not play in a bowl game, but the Tigers did go 8-0. Harvard got far and away the lion's share of national titles that year.
Auburn's 1983 championship is based primarily on what was known as the New York Times Computer Ranking. Ah, yes! Remember back in 1983 cruising the web for national championship selections? Oh, what? You didn't have a computer in 1983? Well, almost no one did, but the New York Times apparently had one that picked a national champion and it wasn't the Miami team of Howard Schnellenberger that defeated "unbeatable" Nebraska. Auburn defeated the Big Ten runner-up, Michigan, 9-3, in the Sugar Bowl.
That 1983 team still does not meet the criteria, though, even with its claim of a national championship. Those Tigers lost a home game to Texas.
In 1993, Auburn was ineligible for a bowl game. Yes, once again a perfect season had been soiled because Auburn had been cheating. The 1957 team was an undefeated national champion that didn't go to a bowl game because of the little matter of buying players. Still, Auburn made its 1993 team national champion – ahead of FSU, which won a No. 1 vs. No. 2 Orange Bowl against Nebraska.
Thus, by our reckoning, the only Auburn team to go undefeated, win a major bowl game, and win the national championship without bought players was the 2010 team led by Cam Newton.