Titus to  3 John

Provisioning the sent.

Provisioning

13See that Zenas the scribe and Apollos are given a good farewell on their journey, that they lack nothing. 14Let our people be taught to do good works in times of emergency that they not be unfruitful. (Titus 3:13-14 BRB)
5Our beloved, in belief you do what you do to the brothers, especially to those who are strangers, 6who have testified concerning your love before the whole assembly for the good things which you have done for them by supplying their needs, as is pleasing to god. 7Because for the sake of his name they have gone forth, taking nothing from the peoples. 3 John 1:5-8 (3 John 1:5-7 BRB)

Paul tells Titus to make sure Zenas and Apollos have what they need for their journey as they leave Crete. Similarly, Gaius, in Third John, is praised by the elder for the way he received, and then sent with provision, some traveling brothers.

The tenses of these passages work well together. In Titus Paul anticipates an eventual departure of Zenas and Apollos, so he reminds Titus to make sure they have what they need before they go. In Third John the brothers have already left Gaius, who had given them provision, so the elder praises him for his actions.

The Book Chain may identify who the brothers were that visited Gaius. Can you think of a pair of brothers who travel around for Jesus and tend to leave a place with provision? Zenas and Apollos come to mind. They left Crete for somewhere with provision. Maybe they went to Gaius. Not hard to imagine if difficult to prove.

Titus mentions that people should be taught to do good works in times of emergency. This statement seems odd given the way it is saddled with the exhortation to provision Zenas and Apollos. What was the emergency? Paul does not specify anything going wrong. In Third John the good work that happened was the way Gaius provisioned the brothers. The emergency in Third John is the way Diotrephes did not receive the brothers and even kicked out of the church people who were in favor of receiving the brothers. Likely, the provisions the brothers needed were tied up in the church, so the good work in time of emergency was Gaius' ability to personally cover their needs.

Slander

7In everything show yourself an example in all good works, and in your teaching let your word be sound. 8Choose sound words that are instructive so that no man can point the finger of scorn at us and so that he who is against us may be shamed when he can find nothing evil to say about us. (Titus 2:7-8 BRB)
9I would have written to the assembly, but he who loves to have preeminence among them FA , their Diotrephes, would not receive us. 10Therefore if I come I will mention the things which he did, gossiping against us with malicious words. Not content with this, he not only did not receive the brothers, but also forbid those who would like to receive them and cast them out of the assembly. 3 John 1:9-10 (3 John 1:9-10 BRB)

Titus and Third John share a theme of slander.

The passages do not use the word slander, but the idea of talking someone down in ways that are not factual is clearly in sight. Paul warns Titus to be careful about his choice of words in teaching so as not to give someone intending harm an easy soundbite that could be twisted. In Third John Diotrephes was "gossiping maliciously" about the elder and others. That means he was saying things that were not true.

Shun

10After you have admonished the heretic 1 time or 2 avoid him 11knowing that he who is such is corrupt. He sins and condemns himself. (Titus 3:10-11 BRB)
9I would have written to the assembly, but he who loves to have preeminence among them FA , their Diotrephes, would not receive us. 10Therefore if I come I will mention the things which he did, gossiping against us with malicious words. Not content with this, he not only did not receive the brothers, but also forbid those who would like to receive them and cast them out of the assembly. 3 John 1:9-10 (3 John 1:9-10 BRB)

Titus is told to shun the heretic after attempting to correct him a time or two. In Third John Diotrephes was wrongly shunning people.

Diotrephes did not receive the brothers that Gaius hosted and sent on their way with provision. And, to make matters worse, he cast out of the church those who wanted to receive the brothers. So Diotrephes was shunning many people. It's an opposit of sorts to what Titus was told, since Diotrephes was shunning people who were not heretics. The elder is clear that Gaius was in the truth and right to receive the brothers, so Diotrephes was wrong to reject them and to cast out those who wanted to receive them. Thus, Diotrephes was working against the truth, and might be the heretic that was to be shunned if he did not change after several attempts at correction.