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Dominating cycles in regular 3-connected graphs

1992
*
Discrete Mathematics
*

Zhu,

doi:10.1016/0012-365x(92)90051-g
fatcat:3clo6irqrzafflsidmk4rikwde
*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. ...##
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On 2-regular subgraphs in polyhedral graphs

2002
*
Discrete Mathematics
*

We show that every polyhedral

doi:10.1016/s0012-365x(01)00329-6
fatcat:k5b2dbtzvzd77d4hfqqms4x7pq
*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. ...

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

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. ...

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

2020
*
Electronic Journal of Combinatorics
*

The disproved Nash-Williams conjecture states that every 4-

doi:10.37236/5505
fatcat:5bjonwmgjjdjzcjmzaoazgzzjy
*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]

2015
*
arXiv
*
pre-print

The disproved Nash Williams conjecture states that every 4-

arXiv:1305.3951v3
fatcat:3dh6yeg44bfbrfebs6o3gppn4m
*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

1993
*
Discrete Mathematics
*

Volkmann, Some upper bounds for the product of the

doi:10.1016/0012-365x(93)90074-4
fatcat:farl2ucjzffohf6kilp5ujsvsa
*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]

2014
*
arXiv
*
pre-print

We present a construction which shows that there is an infinite set of cyclically 4-edge

arXiv:1309.3870v2
fatcat:sq3f6ahhxvdllgiigf6i3ajpoq
*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. ...##
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Complexity of the Minimum Single Dominating Cycle Problem for Graph Classes

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 . ...

##
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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

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

2002
*
Discrete Mathematics
*

We survey results and open problems

doi:10.1016/s0012-365x(01)00325-9
fatcat:wcggfjuntbgrjmegqldzdz55hy
*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

doi:10.47443/cm.2020.0045
fatcat:fg6akhuhj5e3zfqokrl42mbrtq
*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. ...##
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Total domination versus paired-domination in regular graphs

2018
*
Discussiones Mathematicae Graph Theory
*

For k ≥ 2, let G be a

doi:10.7151/dmgt.2026
fatcat:i3hlxf6b2zhxtdv5srjqx3sjhi
*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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