Affiliation:
1. Università della Svizzera italiana, Via Giuseppe Buffi 13, 6900 Lugano, Switzerland
2. Budapest University of Technology and Economics, Műegyetem rakpart 3, 1111 Budapest, Hungary
3. University of Cambridge, Wilberforce Road, CB3 0WB Cambridge, UK
Abstract
A 𝑞-graph with 𝑒 edges and 𝑛 vertices is defined as an 𝑒 × 𝑛 matrix with entries from {0, … , 𝑞}, such that each row of the matrix (called a 𝑞-edge) contains exactly two nonzero entries. If 𝐻 is a 𝑞-graph, then 𝐻 is said to contain an 𝑠-copy of the ordinary graph 𝐹, if a set 𝑆 of 𝑞-edges can be selected from 𝐻 such that their intersection graph is isomorphic to 𝐹, and for any vertex 𝑣 of 𝑆 and any two incident edges 𝑒, 𝑓 ∈ 𝑆 the sum of the entries of 𝑒 and 𝑓 is at least 𝑠. The extremal number ex(𝑛, 𝐹, 𝑞, 𝑠) is defined as the maximal number of edges in an 𝑛-vertex 𝑞-graph such that it does not contain contain an 𝑠-copy of the forbidden graph 𝐹.In the present paper, we reduce the problem of finding ex(𝑛, 𝐹, 𝑞, 𝑞 + 1) for even 𝑞 to the case 𝑞 = 2, and determine the asymptotics of ex(𝑛, 𝐶2𝑘+1, 𝑞, 𝑞 + 1).
Reference8 articles.
1. [1]P. Erdős and M. Simonovits. A limit theorem in graph theory. Studia Sci. Math. Hung. Acad., 1:51-57, 1966.
2. [2]P. Erdős and A. H. Stone. On the structure of linear graphs. Bull. Am. Math. Soc., 52:1087-1091, 1946.
3. [3]Z. Füredi and M. Simonovits. The history of degenerate (bipartite) extremal graph problems. Bolyai Soc. Studies (The Erdös Centennial), 25:167-262, 2013.
4. [4]R. Häggkvist and C. Thomassen. On pancyclic digraphs. J. Combin. Theory Ser. B, 20:20-40, 1976.
5. [5]B. Patkós, Zs. Tuza, and M. Vizer. Extremal graph theoretic questions for q-ary vectors. arXiv preprintarXiv:2305.01919v1 [math.CO], 2022.