A brief summary:

 Today most people believe that Prince Dadian's "masterpieces" were staged, either pre-arranged or simply bought and paid for. During Dadian's life this accusation wasn't so publicly accepted or expressed, but it's not clear whether this was so out of fear of reprisal or because it was slanderous and simply wasn't true.  The public's opinions of Dadian were colored by the press, usually comprised of poor chess writers, who benefited from his "generosity." It's equally true that the ones who gave life to the accusation had personal conflicts with Dadian. So the problem is how to ascertain the truth. First it must be pointed out that there were other less damning but more provable accusations. Dadian was criticized for only sending in his more brilliant wins for publication without mentioning his losses, giving the false impression that he seldom lost or that his opponent was less adept. Secondly, some of his "brilliant" games were shown to be less sound after careful analysis.

   Most of the allegations of pre-arranged games occurred after the incident with Tschigorin in Monte Carlo in 1903, but they all seemed to have had their roots in a separate series of incidents that occurred in Kiev in 1902 and stemmed from the more provable accusations listed above. Dadian had played a series of 12 games with Fyodor Ivanovich Duz-Khotimirsky, a strong local player, winning only 3. One of these wins was particularly charming and Dadian sent that game to La Strategie for publication - without mentioning the details of the series. Duz-Khotimirsky (or the chess club itself), who felt he was treated unfairly, chose his best win in that series for publication. Dadian construed this action as an insult and challenged the officers of the chess club, presumably Duz-Khotimirsky too, to a duel (which never occurred). Two days later, Tschigorin arrived in Kiev for a tournament that was about to take place. Dadian invited Tschigorin to his home, but Tschigorin, aware of the recent incident and either afraid or wishing to avoid any complications, declined the Prince's invitation. Dadian took this as a great insult. Tschigorin  had, in the course of things, annotated some of Dadian's games in his chess column without the deep praise Dadian was accustomed to receiving.

The following year when Tschigorin arrived at Monte Carlo to play in the tournament for which he had been invited, Dadian, who presided over the tournament, threatened to bow out if Tschigorin were permitted to play. The committee acquiesced to Dadian's demand and barred Tschigorin from participating. Although Dadian compensated Tschigorin for his time and expenses, Tschigorin was rightfully upset. Some of his most damaging accusations seemed to have emerged after this incident. 

WilhelmThe2nd notes -

In Panov's book he points out that Dadian often stood out in the contemporary magazines and books his games appeared in because he always seemed to win (and win brilliantly) and because he didn't have any tournament or match victories of significance to his credit. Panov illustrates this by the example of Schiffers book A Chess Self-Teacher which has illustrative games from the great tournament and match players of past and present and two games by Prince Dadian. Panov portrays Schiffers as being in ill-health and going along with Dadian's falsifying games in order to get money to pay his doctors bills. The only factual corroboration I know of is that Schiffers was so sick in the period of 1901-1902 that Chigorin had to take over his column in the magazine Niva afterwards Schiffers resumed the column until his death.

and

“Prince Dadian, of Mingrelia, has a world-wide reputation for doing startling things. The Field, London, calls him ‘the Greco of our day.’ [re: Sicard game] (The Literary Digest, August 24,1901. Page 240).

Of course, Hoffer was the editor of The Field who made that remark. I think it is important to keep in mind this sort of comment on Dadian when considering his reaction to Tschigorin's annotations to his games. Basically, everyone was praising the Prince to the skies whereas Tschigorin was perceived as being envious or spiteful for criticizing his games. Even Kemeny, who confessed himself sympathetic to Tschigorin, seems to have felt this way in his discussion in American Chess Weekly. Dadian was shown a lot of deference because of his wealth and his title ( Dr. Isaac L. Rice, of Rice Gambit fame, was similarly treated at the same time for the same reasons.) It is interesting to note that Hoffer never really delivered on his charges that Tschigorin had made "disparaging and libelous statements ......in the Russian press about the prince not only as a chess player". As the report in Moskovskie Viedemostie later said the explanation for Tschigorin's exclusion given in La Strategie that charged that Tschigorin had refused an invitation to the Prince's home and had publicly stated he wouldn't play in any tournament organized by the Prince doesn't fit the bill (and doesn't appear to be true in any case).

 

Edward Winter, in Chess Notes #1503 and #1542 offers some information provided by Ken Whyld.

#1503 - "The bought games allegation has been widely accepted for about a century and comes under the category of 'common knowledge'. Naturally there were few such claims made during the Prince's lifetime, although Chigorin had something to that effect in his Novoe Vremya column around 1905. [examination by WilhelmThe2nd of Tschigorin's chess column in Novoe Vremya during that time failed to turn up any such claim]  What gives credence to the allegation is the marked difference of quality between his few games in public events and his brilliancies played behind closed doors. A similar allegation was made about his problem compositions.
...
I have recently gone through about 30 of his Serene Highness's games. He obviously played a good deal against slightly weaker opposition and won many games in good style. His victories were sent to the chess press, often with his own notes, and regularly published. There is a saying to the effect that 'a rich man's jokes are always funny'. In chess publishing terms this can be rephrased 'a rich man's victories are always brilliancies', and few if any of these games would have been printed had they come from a humble source."

#1542 (published by Winter, translation by Bernard Cafferty [courtesy of Ken Whyld]) - "In 1902 I once again saw Chigorin; he had been invited to Kiev for exhibitions, which were very successful. Unfortunately his stay in Kiev was marked by a conflict between Chigorin and the Kiev resident Prince Dadian-Mingrelsky, who had a high opinion of himself, thinking himself to be a chess genius. Not long before Chigorin's arrival there was an incident between the Kiev Chess Circle and his 'Serene Highness' the prince. The chess section of a paper edited by the circle printed a game lost by the prince. For this 'insult' the prince officially challenged all the members of the circle to a duel, which did not take place of course, and he tried to get me to his flat to beat me up.
Prince Dadian-Mingrelsky often invited well-known players to his place, and for a suitable fee came to an agreement with them that he would win a 'brilliant' game. Then he would publish the game with his own laudatory comments. It was a long-cherished dream to win against Chigorin in this fashion. However, Chigorin, having learned of the incident of the duel and the prince's intentions towards him, refused to pay a visit to Dadian-Mingrelsky. The prince was enraged, and decided to take his revenge."
(translated from Duz-Khotimirsky. Izbranniye partii, Moscow, 1953. Page 14)

 

BCM , April,1903. Page 163

Prince Dadian of Mingrelia asks us to announce that he is not taking part in the match Kiev vs. St.Petersburg, and that he has no knowledge of the locale of the Kiev Club.
 

BCM, December,1903. Page 516

Prince Dadian of Mingrelia , who lives at Kieff, but is not a member of the local club, did not give any prize in the recent tournament, but he gave a grand banquet, with music, to several chess masters at his house, and also organized consultation games with important prizes.