A note on the upper bound and girth pair of (k;g)-cages
نویسندگان
چکیده
A (k; g)-cage is a k-regular graph of girth g with minimum order. In this work, for all k ≥ 2 and g ≥ 5 odd, we present an upper bound of the order of a (k; g + 1)-cage, which depends on the order of a (k; g)-cage, improving a previous result of Sauer of 1967. We also show that every (k; 11)-cage contains a cycle of length 12, confirming a case of a conjecture of Harary and Kóvacs of 1983.
منابع مشابه
A Note on the Edge-Connectivity of Cages
A (k; g)-graph is a k-regular graph with girth g. A (k; g)-cage is a (k; g)-graph with the smallest possible number of vertices. In this paper we prove that (k; g)cages are k-edge-connected if k ≥ 3 and g is odd.
متن کاملThe distinguishing chromatic number of bipartite graphs of girth at least six
The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The distinguishing chromatic number $chi_{D}(G)$ of $G$ is defined similarly, where, in addition, $f$ is assumed to be a proper labeling. We prove that if $G$ is a bipartite graph of girth at least six with the maximum ...
متن کاملThe second geometric-arithmetic index for trees and unicyclic graphs
Let $G$ be a finite and simple graph with edge set $E(G)$. The second geometric-arithmetic index is defined as $GA_2(G)=sum_{uvin E(G)}frac{2sqrt{n_un_v}}{n_u+n_v}$, where $n_u$ denotes the number of vertices in $G$ lying closer to $u$ than to $v$. In this paper we find a sharp upper bound for $GA_2(T)$, where $T$ is tree, in terms of the order and maximum degree o...
متن کاملImproved Lower Bounds for the Orders of Even Girth Cages
The well-known Moore bound M(k, g) serves as a universal lower bound for the order of k-regular graphs of girth g. The excess e of a k-regular graph G of girth g and order n is the difference between its order n and the corresponding Moore bound, e = n −M(k, g). We find infinite families of parameters (k, g), g > 6 and even, for which we show that the excess of any k-regular graph of girth g is...
متن کاملSmall vertex-transitive graphs of given degree and girth
We investigate the basic interplay between the small k-valent vertex-transitive graphs of girth g and the (k, g)-cages, the smallest k-valent graphs of girth g. We prove the existence of k-valent Cayley graphs of girth g for every pair of parameters k ≥ 2 and g ≥ 3, improve the lower bounds on the order of the smallest (k, g) vertex-transitive graphs for certain families with prime power girth,...
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:
- Discrete Applied Mathematics
دوره 161 شماره
صفحات -
تاریخ انتشار 2013