The total interval number of a graph, I: Fundamental classes

The Total Interval Number of a Graph II: Trees and Complexity

“…For the latter three classes, theyc onjectured that the upper bounds would still hold when the ''2-connected''or' 'triangle-free''restrictions were removed. In [7], we provedthese conjectures for outerplanar and general graphs on n vertices, and we also provedthe Aigner-Andreae conjecture that that max I (G) =(5m + 2)/4 if G is a connected graph with m edges. The proof of their conjecture for planar graphs is quite lengthya nd will appear in a later paper in this series.…”

“…Ac loser examination of the algorithm for trees yields a characterization of the trees requiring m + t intervals for fixed t (Section 4). This is turn yields short proofs of the AignerAndreae extremal bound for trees and the extremal bound in [7] for connected graphs (Section 5).…”

“…The proof of their conjecture for planar graphs is quite lengthya nd will appear in a later paper in this series. Other papers will study the maximum total interval number for cacti or Husimi trees on n vertices, and for connected graphs with m edges having lower bounds on minimum vertexd egree, connectivity,o re dgeconnectivity.M ost of this work appeared in the dissertation of the first author,accepted in 1987 [6].…”

Supereulerian graphs: A survey

“…Extending results of Andreae and Aigner [1] (who did the triangle-free case), Kostochka [91] and Kratzke and West ([93], [94]) obtained various results, including: Theorem 6.7 [93], [94] If G is a planar graph of order n ≥ 3 then I(G) ≤ 2n − 3. 2 Theorem 6.8 [91], [94] If G is a connected graph with m ≥ 2 edges, then…”

“…We say that {v 1 , v 2 , ...., v n } is a representation of G. Griggs and West [78] and Andreae and Aigner [1] defined the total interval number I(G) to be the minimum value of n i=1 |v i |, among all representations of G. (The interval number of G, defined by Trotter and Harary [135], is different: it is the minimum of max 1≤i≤n |v i | over all representations of G.) Andreae and Aigner [1] obtained an upper bound on I(G) in terms of dominating trails, and Kratzke and West [93,94] noted that equality holds if G is triangle-free: Lemma 6.5 [1], [93], [94] Let k be a nonnegative integer. If a connected graph G is at most k edges short of having a dominating trail, then…”