Regular 4‐critical graphs of even degree

Abstract

In 1960, Dirac posed the conjecture that r-connected 4-critical graphs exist for every r ! 3. In 1989, Erdo´s conjectured that for every r ! 3 there exist r-regular 4-critical graphs. In this paper, a technique of constructing r-regular r-connected vertex-transitive 4-critical graphs for even r ! 4 is presented. Such graphs are found for r ¼ 6, 8, 10. ß

Erdős regular graphs of even degree

“…It follows from the next lemma that every normal 4-chromatic circulant is 4-critical. Lemmas 1-3 have been proved in [6,7,21]. Lemma 1.…”

“…There are several known sufficient conditions for a 3-chromatic circulant to be periodic. They are collected in the following lemma proved in [7].…”

“…More detailed information on critical graphs and related topics can be found in the book [15]. Examples of regular 4-critical graphs of degree 4, 6, 8 and 10 have been recently reported in [5,6,7,21]. In this paper, we describe results of a computer search for graphs of degree r = 12, 14, 16 that are both Erdős and Dirac graphs.…”

“…In 1989, Erdős conjectured that for every r ≥ 3 there exist r-regular 4-critical graphs. A method for finding r-regular 4-critical graphs and the numbers of such graphs for r ≤ 10 have been reported in [6,7]. Results of a computer search for graphs of degree r = 12, 14, 16 are presented.…”

Computing Wiener Index of C24n Fullerenes

“…1 Most of the results on Wiener index are obtained by working on other features of the graph; for more details about Wiener index see Refs. [2][3][4][5].…”