2 research outputs found

    Simple cubic graphs with no short traveling salesman tour

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    Let tsp(G)tsp(G) denote the length of a shortest travelling salesman tour in a graph GG. We prove that for any ε>0\varepsilon>0, there exists a simple 22-connected planar cubic graph G1G_1 such that tsp(G1)(1.25ε)V(G1)tsp(G_1)\ge (1.25-\varepsilon)\cdot|V(G_1)|, a simple 22-connected bipartite cubic graph G2G_2 such that tsp(G2)(1.2ε)V(G2)tsp(G_2)\ge (1.2-\varepsilon)\cdot|V(G_2)|, and a simple 33-connected cubic graph G3G_3 such that tsp(G3)(1.125ε)V(G3)tsp(G_3)\ge (1.125-\varepsilon)\cdot|V(G_3)|

    Weak oddness as an approximation of oddness and resistance in cubic graphs

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    We introduce weak oddness ωw\omega_{\textrm w}, a new measure of uncolourability of cubic graphs, defined as the least number of odd components in an even factor. For every bridgeless cubic graph GG, ρ(G)ωw(G)ω(G)\rho(G)\le\omega_{\textrm w}(G)\le\omega(G), where ρ(G)\rho(G) denotes the resistance of GG and ω(G)\omega(G) denotes the oddness of GG, so this new measure is an approximation of both oddness and resistance. We demonstrate that there are graphs GG satisfying ρ(G)<ωw(G)<ω(G)\rho(G) < \omega_{\textrm w}(G) < \omega(G), and that the difference between any two of those three measures can be arbitrarily large. The construction implies that if we replace a vertex of a cubic graph with a triangle, then its oddness can decrease by an arbitrarily large amount
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