Coloring vertices and edges of a graph by nonempty subsets of a set

New classes of set-sequential trees

“…The motivation behind Conjecture 1.2 presented in [6] is easily applied to show that the conjecture, if true, would imply the existence of a finite set of caterpillars of some given diameter k for which if these caterpillars are set-sequential, then all caterpillars of diameter k with only odd-degree vertices are set-sequential. This statement is not extended to caterpillars with even-degree vertices because infinite classes of such caterpillars that are not set-sequential are shown to exist in [5].…”

“…Mehta and Vijayakumar [4] showed that all paths of length 2 n−1 are set-sequential if n ≥ 5. Hegde [5] showed that no graph with exactly 00001 00111 = 0v 7 = 0v 8 01101 = 0v 1 00010 0v 3 = 0v 4 = 0v 5 1q 8 1q 7 Figure 1: An illustration of the method for generating set-sequential trees introduced in [6] that was presented as motivation for Conjecture 1. 2.…”

“…Balister, Győri, and Schelp [6] proposed a general method to generate set-sequential trees by adding 2 n−1 pendant edges to a set-sequential tree with 2 n−1 vertices. Specifically, consider a set-sequential tree T with 2 n−1 vertices.…”

“…It was conjectured in [6] that for any choice of the v i 's satisfying (1), there always exists a choice of the p i 's and q i 's for which the two conditions above hold, as formalized below.…”

“…, v 2 n−1 or the number of distinct vectors v i is restricted. We then show that all caterpillars T with only odd-degree vertices of diameter k such that k ≤ 18 or |V (T )| ≥ 2 k−1 are set-sequential, assuming that T has 2 n−1 vertices for some positive integer n. We also present a new method of constructing a set-sequential tree by connecting four copies of an existing set-sequential tree, which enables us to prove the set-sequentialness of a broader class of trees that is not covered by the method presented in [6].…”