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
The chromaticity of certain graphs with five triangles
β Scribed by Nian-Zu Li; Earl Glen Whitehead Jr
- Publisher
- Elsevier Science
- Year
- 1993
- Tongue
- English
- Weight
- 397 KB
- Volume
- 122
- Category
- Article
- ISSN
- 0012-365X
No coin nor oath required. For personal study only.
β¦ Synopsis
Let W(n, k) denote the graph of order n obtained from the wheel I+', by deleting all but k consecutive spokes. In this note, we study the chromaticity of graphs which share certain properties of U'(n, 6) which can be obtained from the coeffictents of the chromatic polynomial of W(n, 6). In particular, we prove that W(n,6) is chromatically unique for all integers n>X. We also obtain two additional families of chromatically unique graphs.
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Let Kr~ be the complete graph on N vertices, and assume that each edge is assigned precisly one of three possible colors. An old and difficult problem is to find the minimum number of monochromatic triangles as a function of N. We are not able to solve this problem, but we can give sharp bounds for
## Abstract Let __m__ and __n__ be nonnegative integers. Denote by __P__(__m,n__) the set of all triangleβfree graphs __G__ such that for any independent __m__βsubset __M__ and any __n__βsubset __N__ of __V__(__G__) with __M__ β© __N__ = Γ, there exists a unique vertex of __G__ that is adjacent to e