Highly connected counterexamples to a conjecture on ά -domination

نویسنده

  • Zsolt Tuza
چکیده

An infinite class of counterexamples is given to a conjecture of Dahme et al. [Discuss. Math. Graph Theory, 24 (2004) 423–430.] concerning the minimum size of a dominating vertex set that contains at least a prescribed proportion of the neighbors of each vertex not belonging to the set.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

On the diameter of domination bicritical graphs

For a graph G, we let γ(G) denote the domination number of G. A graph G is said to be k-bicritical if γ(G) = k and γ(G − {x, y}) < k for any two vertices x, y ∈ V (G). Brigham et al. [Discrete Math. 305 (2005), 18–32] conjectured that the diameter of a connected k-bicritical graph is at most k − 1. However, in [Australas. J. Combin. 53 (2012), 53–65], counterexamples of the conjecture for k = 4...

متن کامل

Counterexamples to the Cubic Graph Domination Conjecture

Let v(G) and γ(G) denote the number of vertices and the domination number of a graph G, respectively, and let ρ(G) = γ(G)/v(G). In 1996 B. Reed conjectured that ifG is a cubic graph, then γ(G) ≤ ⌈v(G)/3⌉. In 2005 A. Kostochka and B. Stodolsky disproved this conjecture for cubic graphs of connectivity one and maintained that the conjecture may still be true for cubic 2-connected graphs. Their mi...

متن کامل

On the connectivity of minimum and minimal counterexamples to Hadwiger's Conjecture

The main result of this paper is the following: Any minimal counterexample to Hadwiger’s Conjecture for the k-chromatic case is 2k 27 -connected. This improves the previous known bound due to Mader [W. Mader, Über trennende Eckenmengen in homomorphiekritischen Graphen, Math. Ann. 175 (1968) 243–252], which says that any minimal counterexample to Hadwiger’s Conjecture for the k-chromatic case is...

متن کامل

On the ratio of the domination number and the independent domination number in graphs

We let γ(G) and i(G) denote the domination number and the independent domination number ofG, respectively. Recently, Rad and Volkmann conjectured that i(G)/γ(G) ≤ ∆(G)/2 for every graph G, where ∆(G) is the maximum degree of G. In this note, we construct counterexamples of the conjecture for ∆(G) ≥ 6, and give a sharp upper bound of the ratio i(G)/γ(G) by using the maximum degree of G.

متن کامل

A note on Fouquet-Vanherpe’s question and Fulkerson conjecture

‎The excessive index of a bridgeless cubic graph $G$ is the least integer $k$‎, ‎such that $G$ can be covered by $k$ perfect matchings‎. ‎An equivalent form of Fulkerson conjecture (due to Berge) is that every bridgeless‎ ‎cubic graph has excessive index at most five‎. ‎Clearly‎, ‎Petersen graph is a cyclically 4-edge-connected snark with excessive index at least 5‎, ‎so Fouquet and Vanherpe as...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:
  • Discussiones Mathematicae Graph Theory

دوره 25  شماره 

صفحات  -

تاریخ انتشار 2005