## Abstract In 1968, Vizing [Uaspekhi Mat Nauk 23 (1968) 117β134; Russian Math Surveys 23 (1968), 125β142] conjectured that for any edge chromatic critical graph ${{G}} = ({{V}}, {{E}})$ with maximum degree $\Delta$, $|{{E}}| \geq {{{1}}\over {{2}}}\{(\Delta {{- 1}})|{{V}}| + {{3}}\}$. This conject
The maximum size of graphs satisfying a degree condition
β Scribed by Yiping Qiu; Xiao Feng Jia
- Publisher
- Elsevier Science
- Year
- 1992
- Tongue
- English
- Weight
- 196 KB
- Volume
- 104
- Category
- Article
- ISSN
- 0012-365X
No coin nor oath required. For personal study only.
β¦ Synopsis
Let G be a simple graph of order n without isolated vertices. If the integer h satisfies
In this note the maximum size of Sri(h)--graphs is determined. A result of Krol and Veldman on critically h-connected graphs follows as a corollary.
π SIMILAR VOLUMES
Graphs with n + k vertices in which every set of n +j vertices induce a subgraph of maximum degree at least n are considered. For j = 1 and for k fairly small compared to n, we determine the minimum number of edges in such graphs.
## Abstract A total dominating set, __S__, in a graph, __G__, has the property that every vertex in __G__ is adjacent to a vertex in __S__. The total dominating number, Ξ³~__t__~(__G__) of a graph __G__ is the size of a minimum total dominating set in __G__. Let __G__ be a graph with no component of
Frydrych, W., All nonhamiltonian tough graphs satisfying a 3-degree sum and Fan-type conditions, Discrete Mathematics 121 (1993) 93-104. It is shown that if G is a l-tough nonhamiltonian graph on even number vertices n>4 such that d(x)+d(y)+d(z)>n for every triple of mutually distinct and nonadjace
It is well known that a graph G of orderp 2 3 is Hamilton-connected if d(u) +d(v) 2 p + 1 for each pair of nonadjacent vertices u and w. In this paper we consider connected graphs G of order at least 3 for which where N ( z ) denote the neighborhood of a vertex z. We prove that a graph G satisfying
A vertex x in a subset X of vertices of an undirected graph is redundant if its dosed neighborhood is contained in the union of closed neighborhoods of vertices of X-{x}. In the context of a communications network, this means that any vertex that may receive communications from X may also be irdorme