The Number of Graphs Not Containing K3, 3 as a Minor
نویسندگان
چکیده
We derive precise asymptotic estimates for the number of labelled graphs not containing K3,3 as a minor, and also for those which are edge maximal. Additionally, we establish limit laws for parameters in random K3,3-minor-free graphs, like the number of edges. To establish these results, we translate a decomposition for the corresponding graphs into equations for generating functions and use singularity analysis. We also find a precise estimate for the number of graphs not containing the graph K3,3 plus an edge as a minor.
منابع مشابه
Interval minors of complete bipartite graphs
Interval minors of bipartite graphs were recently introduced by Jacob Fox in the study of Stanley-Wilf limits. We investigate the maximum number of edges in Kr,s-interval minor free bipartite graphs. We determine exact values when r = 2 and describe the extremal graphs. For r = 3, lower and upper bounds are given and the structure of K3,s-interval minor free graphs is studied.
متن کاملDense graphs have K3, t minors
Let K ∗ 3,t denote the graph obtained from K3,t by adding all edges between the three vertices of degree t in it. We prove that for each t ≥ 6300 and n ≥ t + 3, each n-vertex graph G with e(G) > 1 2 (t + 3)(n− 2)+ 1 has a K ∗ 3,t -minor. The bound is sharp in the sense that for every t , there are infinitely many graphs Gwith e(G) = 2 (t+ 3)(|V (G)|− 2)+ 1 that have no K3,t -minor. The result c...
متن کاملThe circumference of a graph with no K3, t-minor
The class of graphs with no K3,t-minors, t ≥ 3, contains all planar graphs and plays an important role in graph minor theory. In 1992, Seymour and Thomas conjectured the existence of a function α(t) > 0 and a constant β > 0, such that every 3-connected n-vertex graph with no K3,t-minors, t ≥ 3, contains a cycle of length at least α(t)n . The purpose of this paper is to confirm this conjecture w...
متن کاملGraphs containing triangles are not 3-common
Jagger, S̆t̆ov́ıc̆ek and Thomason [3] defined the class of k-common graphs, and showed among other results that every graph containing K4 as a subgraph is not 2-common. We prove that every graph containing K3 as a subgraph is not 3-common.
متن کاملA Characterization of Graphs with No Octahedron Minor
It is proved that a graph does not contain an octahedron minor if and only if it is constructed from {K1,K2,K3,K4}∪{C 2 2n−1 : n ≥ 3} and five other internally 4-connected graphs by 0-, 1-, 2-, and 3-sums.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Electr. J. Comb.
دوره 15 شماره
صفحات -
تاریخ انتشار 2008