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### Dominating cycles in regular 3-connected graphs

Bill Jackson, Hao Li, Yongjin Zhu
1992 Discrete Mathematics
Zhu, Dominating cycles in regular 3-connected graphs, Discrete Mathematics 102 (1992) 163-176. Let G be a 3-connected, k-regular graph on at most 4k vertices.  ...  We show that, for k > 63, every longest cycle of G is a dominating cycle. We conjecture that G is in fact hamiltonian.  ...  Let G be a 3-connected, k-regular graph on at most 4k vertices. Then for k 2 63, every longest cycle in G is a dominating cycle. Our result is closely related to the work of H.A. Jung.  ...

### On 2-regular subgraphs in polyhedral graphs

Frank Göring
2002 Discrete Mathematics
We show that every polyhedral graph G contains a 2-regular subgraph U such that G − U is a forest of trees with at most three leaves.  ...  With this tool we give a partial solution of a problem posed in a problem session at the Workshop "Cycles and Colourings '98" in Starà a Lesnà a, Slovakia.  ...  Now connect each of the red vertices with the green vertex in the incident hexagon. The result is a 3-connected plane graph G without a 1-dominating 2-regular subgraph . Example 2.  ...

### On the Connected End Equitable Domination of Graphs

2016 International Journal of Scientific and Innovative Mathematical Research
In this paper we introduce the connected end equitable domination of graphs.  ...  The connected domination number is the minimum size of such a set. An equitable dominating set in a graph is called connected equitable dominating set in the subgraph induced by is connected.  ...  Example 2 . 23 . 223 Let be a cycle with three attached edges in each vertex as inFigure 3. Fig. 3 : 3 Fig. 3: Fig. 4 : 4 helm graph Proposition2.26.  ...

### Connected, regular and split liar domination on fuzzy graphs

Roseline Mary S., Ruban Raj S., Maria Joseph J.
2020 Malaya Journal of Matematik
In this paper we discussed Connected, Regular and Split liar domination on fuzzy graphs and also discussed some of their properties.  ...  Liar domination set in a fuzzy graph is the set to identify the intruder location in a computer network / communication network with minimum fuzzy cardinality of the nodes.  ...  In this paper we discussed Connected, Regular, Split liar domination on fuzzy graphs and their properties.  ...

### The Nash-Williams Conjecture and the Dominating Cycle Conjecture

Arthur Hoffmann-Ostenhof
2020 Electronic Journal of Combinatorics
The disproved Nash-Williams conjecture states that every 4-regular 4-connected graph has a hamiltonian cycle.  ...  We show that a modification of this conjecture is equivalent to the Dominating Cycle Conjecture.  ...  Nash-Williams Conjecture (NWC): Every 4-regular 4-connected graph has a hamiltonian cycle. Dominating Cycle Conjecture (DCC): Every cyclically 4-edge connected cubic graph has a dominating cycle.  ...

### Nash Williams Conjecture and the Dominating Cycle Conjecture [article]

Arthur Hoffmann-Ostenhof
2015 arXiv   pre-print
The disproved Nash Williams conjecture states that every 4-regular 4-connected graph has a hamiltonian cycle.  ...  We show that a modification of this conjecture is equivalent to the Dominating Cycle Conjecture.  ...  Nash Williams Conjecture (NWC): Every 4-regular 4-connected graph has a hamiltonian cycle. Dominating Cycle Conjecture (DCC): Every cyclically 4-edge connected cubic graph has a dominating cycle.  ...

### Some upper bounds for the product of the domination number and the chromatic number of a graph

Jerzy Topp, Lutz Volkmann
1993 Discrete Mathematics
Volkmann, Some upper bounds for the product of the domination number and the chromatic number of a graph, Discrete Mathematics 118 (1993) 2899292.  ...  If G is a connected regular graph with n > 6 and G is difirent from the cycle C7, then Proof. If G is a complete graph, then yx = n and (1) holds for n 3 6.  ...  If G is a connected graph with n > 5, then yx <a n2. Theorem B. If G is a regular graph with n 3 5, then yx <ff n2 , In this note we present some improvements of the above inequalities.  ...

### Improved bounds for the shortness coefficient of cyclically 4-edge connected cubic graphs and snarks [article]

Klas Markström
2014 arXiv   pre-print
We present a construction which shows that there is an infinite set of cyclically 4-edge connected cubic graphs on n vertices with no cycle longer than c_4 n for c_4=12/13, and at the same time prove that  ...  The graphs we construct are snarks so we get the same upper bound for the shortness coefficient of snarks, and we prove that the constructed graphs have an oddness growing linearly with the number of vertices  ...  Given a matching M of size 3 in any cyclically 4-edge connected cubic graph on n ≤ 22 vertices there is a dominating cycle containing M . 2.  ...

### Complexity of the Minimum Single Dominating Cycle Problem for Graph Classes

Hiroshi ETO, Hiroyuki KAWAHARA, Eiji MIYANO, Natsuki NONOUE
2018 IEICE transactions on information and systems
dominating problem, graph classes, (in)tractability, (in)approximability  ...  We first show that MinSDC is still NPhard to approximate even when restricted to planar, bipartite, chordal, or r-regular (r ≥ 3).  ...  We first give the reduction for 3-regular graphs, and then modify it to one for general r ≥ 4: Let G be a 3-regular graph, to vertex y where (x, Suppose that H contains a dominating cycle C H .  ...

### Page 5325 of Mathematical Reviews Vol. , Issue 2000h [page]

2000 Mathematical Reviews
We also show that not all cyclically 3- edge connected 3-regular (planar) graphs admit a tree-partition, and present the smallest counterexamples.”  ...  Summary: “If G is a 4-connected maximal planar graph, then G is Hamiltonian (by a theorem of Whitney), implying that its dual graph G* is a cyclically 4-edge connected 3-regular planar graph admitting  ...

### Spanning Star Trees in Regular Graphs

Jerrold W. Grossman
1997 Graphs and Combinatorics
In this case W forms a weakly connected but strongly acyclic dominating set for G.  ...  We prove that for every r ≥ 3, there exist r-regular n-vertex graphs that have spanning star trees, and there exist r-regular n-vertex graphs that do not have spanning star trees, for all n sufficiently  ...  Will adding higher connectivity assumptions renders the problem easier (or, in the context of the results of §3, make it more difficult to find bad graphs)?  ...

### Page 658 of Mathematical Reviews Vol. , Issue 95b [page]

1995 Mathematical Reviews
95b:05121 05 95b:05121 05C38 05C45 Fleischner, Herbert (A-OAW-I; Vienna) Uniqueness of maximal dominating cycles in 3-regular graphs and of Hamiltonian cycles in 4-regular graphs.  ...  Graph Theory 18 (1994), no. 5, 449-459. The author constructs 3-regular graphs G having a dominating cycle C for which there is no other cycle C,; with V(C) C V(C;).  ...

### On some intriguing problems in hamiltonian graph theory—a survey

H.J. Broersma
2002 Discrete Mathematics
We survey results and open problems in hamiltonian graph theory centered around three themes: regular graphs, t-tough graphs, and claw-free graphs.  ...  A cycle C of a graph G is called a dominating cycle if V (G)\V (C) is an independent set of G. Theorem 2.4 (Jackson et al. [46] ). Let G be a 3-connected k-regular graph on at most 4k vertices.  ...  If G contains a dominating cycle; then G is hamiltonian. Theorem 2.9 (Broersma et al. [16] ). Let G be a 2-connected k-regular graph on at most 4k − 3 vertices.  ...

### Total dominating functions of graphs: antiregularity versus regularity

2020 Contributions to Mathematics
A set S of vertices in a nontrivial connected graph G is a total dominating set if every vertex of G is adjacent to some vertex of S.  ...  We present some results dealing with properties of regular total dominating functions of graphs. In particular, regular total dominating functions of trees are investigated.  ...  Furthermore, we thank the anonymous referees whose valuable suggestions resulted in an improved paper.  ...

### Total domination versus paired-domination in regular graphs

Joanna Cyman, Magda Dettlaff, Michael .A. Henning, Magdalena Lemańska, Joanna Raczek
2018 Discussiones Mathematicae Graph Theory
For k ≥ 2, let G be a connected k-regular graph. It is known [Schaudt, Total domination versus paired domination, Discuss. Math. Graph Theory 32 (2012) 435-447] that γ pr (G)/γ t (G) ≤ (2k)/(k + 1).  ...  A subset S of vertices of a graph G is a dominating set of G if every vertex not in S has a neighbor in S, while S is a total dominating set of G if every vertex has a neighbor in S.  ...  If G is a connected cubic graph, then γ pr (G) γ t (G) ≤ 3 2 , with equality if and only if G is the Petersen graph. Total Domination Versus Paired-Domination in Regular Graphs 579 Proof.  ...
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