## Abstract It is shown that the minimum number of vertices in a triangleβfree 5βchromatic graph is at least 19.
Triangle-free four-chromatic graphs
β Scribed by Guoping Jin
- Publisher
- Elsevier Science
- Year
- 1995
- Tongue
- English
- Weight
- 640 KB
- Volume
- 145
- Category
- Article
- ISSN
- 0012-365X
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β¦ Synopsis
For given n, let G be a triangle-free graph of order n with chromatic number at least 4. In this paper, we shall prove a conjecture of H/iggkvist by determining the maximal value of 6(G).
π SIMILAR VOLUMES
We give a new example of a triangle-free =-chromatic graph: the vertices of G form a WX 00 matrix, V(G) = [S,j], i,. i = 1,2, . . . The vertex Ui,j is connected with every vertex of the (i + j)th column. G is triangle-free: if A has the smallest column-index among {A, B, C} c V(G) and AB, ACE E(G),
It follows from the results of , Gyirfis and Lehel (1985), and Kostochka (1988) that 4 ~x\* ## ~5 where x\* = max {X(G): G is a triangle-free circle graph}. We show that X\* ? 5 and thus X\* = 5. This disproves the conjecture of Karapetyan that X\* = 4 and answers negatively a question of Gyirfis
We show that for every k β₯ 1 and Ξ΄ > 0 there exists a constant c > 0 such that, with probability tending to 1 as n β β, a graph chosen uniformly at random among all triangle-free graphs with n vertices and M β₯ cn 3/2 edges can be made bipartite by deleting Ξ΄M edges. As an immediate consequence of th