K-minimal 3-connected cubic graphs
WebWe prove that for every d ≥ 3 there is an infinite family of Hamiltonian 3-connected graphs with minimum degree d, with a bounded number of Hamiltonian cycles. It is shown that if a 3-regular graph G has a unique longest cycle C, at least two components of G − E ( C) have an odd number of vertices on C, and that there exist 3-regular graphs ... WebMay 22, 2014 · In this paper, we propose a related problem as follows: Given a graph G = ( V, E) and a vertex-weight function w, the goal is to find a minimum weight vertex set F ⊆ V such that every connected subgraph on k vertices has at least one vertex from F. Call such a set F as a connected k-subgraph cover ( VCC k) and the problem as a minimum weight ...
K-minimal 3-connected cubic graphs
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WebIn this paper we show that if G≠K(3,3) is a connected cubic graph, then i(G)/γ(G)≤4/3. This answers a question posed in Goddard [6] where the bound of 3/2 is proven. ... 159–162] that every connected graph with minimum degree at least two that is not a cycle on five vertices has a dominating set DD and a total dominating set TT which are ... WebJan 1, 2024 · Merker proves in [1] that for every integer k ≥ 4 there exists a 3-connected cubic planar graph G of circumference at least k which satisfies C (G) ∩ [k, 2k + 1] = ∅. In …
WebSep 2, 2024 · For general 3-connected planar graphs, Merker conjectured that there exists some positive integer $c$ such that $f (k) \le 2k + c$ for any positive integer $k$. We give … WebDec 18, 2024 · We organize a table of regular graphs with minimal diameters and minimal mean path lengths, large bisection widths and high degrees of symmetries, obtained by enumerations on supercomputers. ... Let G be a connected regular graph of N vertices and degree k. The distance d ... McGee graph; (3,7)-Cage; Smallest cubic crossing number-8; …
WebApr 1, 1986 · Minimally k-connected graphs have applications in the field of costminimizing network design (e.g., [6,7,15]). For example, minimally kconnected graphs provide optimal … WebEvery cubic 3-connected 14-Halin graph with a connected core is hamilto- nian. Proof. Let G have k inner cycles (k 6 14) and denote by K the core of G. Then K may only take one of …
WebJul 24, 2024 · The smallest 3-connected cubic graph containing H'' can be obtained from H'' by adding three edges joining the vertices of degree 2. However, there are many ways how to construct 3-connected cubic graphs containing H'' as a subgraph. For example, let G be any 3-connected cubic graph containing an induced 6-cycle C.
how to use rtk testWebwhich do not have an induced copy of the claw K(1,3): Proposition 3 [6] If G is a noncomplete K(1,3)-free graph G then τ(G) = κ(G)/2. For r ≥ 4 an r-regular r-connected K(1,3)-free graph with large order need not have order a multiple of r. (For example, the line graph of a random (3,4)-biregular graph is 5-regular and almost surely 5 ... how to use rtgenWebApr 30, 2015 · The resulting graph H_1 is 3-connected. For k\ge 2, H_k has connectivity 2. Always H_k has 14k vertices and is cubic. Fig. 1 The graph F Full size image Theorem 2.2 i (H_k)=5k and \gamma (H_k)=4k. Proof First, we prove \gamma (H_k)=4k. Since \ {a^1, b^3, b^4, a^6\} is a dominating set in F, we have \gamma (H_k) \le 4k. how to use rtkWebFeb 3, 2024 · 3-connected cubic graphs cycles ccby-4.0 Photo by patricia serna on Unsplash Editorial introduction Read article at ArXiv The existence of cycles of a prescribed length … organizing containers for bedroomWebThe complete bipartite graph K3,3 is an example of a bicubic graph In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words a cubic graph is a 3- regular graph. Cubic graphs are also called trivalent graphs. A bicubic graph is a cubic bipartite graph. Contents [ hide] 1 Symmetry how to use rtk test kitWebBarnette's conjecture states that every cubic bipartite polyhedral graph is Hamiltonian. By Steinitz's theorem, a planar graph represents the edges and vertices of a convex polyhedron if and only if it is polyhedral. A three-dimensional polyhedron has a cubic graph if and only if it is a simple polyhedron . And, a planar graph is bipartite if ... organizing contacts in outlookWebk(G) is the size of a minimum k-conversion set of G. A set X ⊆ V is a decycling set, or feedback vertex set, if and only if G[V − X] is ... study 3-connected cubic graphs in Section 5.2, where we construct classes of 3-connected cubic graphs H m of arbitrary girth (and other properties) ... how to use rth mavic mini