On regular graphs of girth six arising from projective planes
نویسندگان
چکیده
In 1967, Brown constructed small k-regular graphs of girth six as induced subgraphs of the incidence graph of a projective plane of order q, q ≥ k. Examining the construction method, we prove that starting from PG(2, q), q = p, p prime, there are no other constructions using this idea resulting in a (q + 1− t)-regular graph of girth six than the known ones, if t is not too large (t ≤ p and roughly t < q/8). Both algebraic and combinatorial tools are used.
منابع مشابه
Incidence Matrices of Projective Planes and of Some Regular Bipartite Graphs of Girth 6 with Few Vertices
Let q be a prime power and r = 0, 1 . . . , q − 3. Using the Latin squares obtained by multiplying each entry of the addition table of the Galois field of order q by an element distinct from zero, we obtain the incidence matrices of projective planes and the incidence matrices of (q− r)-regular bipartite graphs of girth 6 and q − rq− 1 vertices in each partite set. Moreover, in this work two La...
متن کاملAn Extremal Characterization of Projective Planes
In this article, we prove that amongst all n by n bipartite graphs of girth at least six, where n = q + q + 1 ≥ 157, the incidence graph of a projective plane of order q, when it exists, has the maximum number of cycles of length eight. This characterizes projective planes as the partial planes with the maximum number of quadrilaterals.
متن کاملThe Endomorphism Type of Certain Bipartite Graphs and a Characterization of Projective Planes
In [2] Fan determines the endomorphism type of a finite projective plane. In this note we show that Fan’s result actually characterizes the class of projective planes among the finite bipartite graphs of diameter three. In fact, this will follow from a generalization of Fan’s theorem and its converse to all finite bipartite graphs with diameter d and girth g such that (1) d + 1 < g ≤ 2d, and (2...
متن کاملOn Hamiltonian Regular Graphs
In this paper we shall determine, when 1 = 6, bounds for numbers f(k, I) and F{k, 1) defined as follows: f{k, l)/F(k, I) is defined to be the smallest integer n for which there exists a regular graph/Hamiltonian regular graph of valency k and girth I having n vertices. The problem of determining minimal regular graphs of given girth was first considered by Tutte [9]. Bounds for f(k, I) have bee...
متن کاملSymmetric Cubic Graphs of Girth at Most 7
By a symmetric graph we mean a graph X which automorphism group acts transitively on the arcs of X. A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. Tutte [31, 32] showed that every finite symmetric cubic graph is s-regular for some s ≤ 5. It is well-known that there are precisely five symmetric cubic graphs of girth less than 6. All these graphs can be re...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Eur. J. Comb.
دوره 34 شماره
صفحات -
تاریخ انتشار 2013