On connectedness and hamiltonicity of direct graph bundles

A necessary and sufficient condition for connectedness of direct graph bundles is given where the fibers are cycles. We also prove that all connected direct graph bundles $X=C_stimes^{alpha}C_t$ are Hamiltonian

Hamiltonicity of cartesian and direct graph bundle

Ciklična svežnjevska Hamiltonskost cbH(G) grafa G je najmanjši n, za katerega obstaja tak avtomorfizem grafa G, da je kartezični grafovski sveženj, katerega baza je cikel na n točkah in vlakno graf G, Hamiltonov graf. Podamo oceno za cbH(G) in to oceno dokažemo. Podamo potrebne in zadostne pogoje za povezanost direktnih grafovskih svežnjev katerih vlakna so cikli. Pokažemo tudi, da so vsi povezani direktni grafovski svežnji ciklov nad cikli Hamiltonovi grafi.Cyclic bundle Hamiltonicity cbH(G) is the minimal n for which there is an automorphism of G such that the Cartesian graph bundle with cycle as base and G as fiber is Hamiltonian. We prove the value of cbH(G). A necessary and sufficient condition for connectedness of direct graph bundles where the fibers are cycles is given. It is also proved that all connected direct graph bundles cycles over cycles are Hamiltonian

Cyclic bundle Hamiltonicity

Cyclic bundle Hamiltonicity ▫$cbH(G)$▫ of a graph ▫$G$▫ is the minimal ▫$n$▫ for which there is an automorphism ▫$alpha$▫ of ▫$G$▫ such that the graph bundle ▫$C_nBox^{alpha} G$▫ is Hamiltonian. We define ▫$nabla (tilde{G}_{alpha})_{min}$▫, an invariant that is related to the maximal vertex degree of spanning trees suitably involving the symmetries of ▫$G$▫ and prove ▫$cbH(G) leq nabla(tilde{G}_{alpha})_{min} leq cbH(G)+1$▫ for any non-trivial connected graph ▫$G$▫

On L(d, 1)-labelling of trees

Given a graph ▫$G$▫ and a positive integer ▫$d$▫, an ▫$L(d,1)$▫-labelling of ▫$G$▫ is a function ▫$f$▫ that assigns to each vertex of ▫$G$▫ a non-negative integer such that if two vertices ▫$u$▫ and ▫$v$▫ are adjacent, then ▫$|f(u)-f(v) |ge d$▫ and if ▫$u$▫ and ▫$v$▫ are at distance two, then ▫$|f(u)-f(v)| ge 1$▫. The ▫$L(d,1)$▫-number of ▫$G$▫, ▫$lambda_d(G)$▫, is the minimum ▫$m$▫ such that there is an ▫$L(d,1)$▫-labelling of ▫$G$▫ with ▫$f(V) subseteq {0,1,dots , m}$▫. A tree ▫$T$▫ is of type 1 if ▫$lambda_d(T) = Delta+d-1$▫ and is of type 2 if ▫$lambda_d(T) ge Delta+d$▫. This paper provides sufficient conditions for ▫$lambda_d(T)=Delta+d-1$▫ generalizing the results of Wang [W. Wang, The ▫$L(2,1)$▫-labeling of trees, Discrete Appl. Math. 154 (2006) 598-603] and Zhai, Lu, and Shu [M. Zhai, C. Lu and J. Shu, A note on ▫$L(2,1)$▫-labeling of Trees, Acta. Math. Appl. Sin. 28 (2012) 395-400] for ▫$L(2,1)$▫-labelling

On connectedness and hamiltonicity of direct graph bundles

A necessary and sufficient condition for connectedness of direct graph bundles where the fibers are cycles is given. It is also proved that all connected direct graph bundles ▫$X=C_stimes^{alpha}C_t$▫ are Hamiltonian

The diameter of strong orientations of strong products of graphs

Let G and H be graphs, and G☒H the strong product of G and H. We prove that for any connected graphs G and H there is a strongly connected orientation D of G☒H such that diam(D) ≤ 2r+15, where r is the radius of G☒H. This improves the general bound diam(D) ≤ 2r2+2r for arbitrary graphs, proved by Chvatal and Thomassen

On connectivity and hamiltonicity of direct graph bundles

A necessary and sufficient condition for connectedness of direct graph bundles where the fibers are cycles is given. It is also proved that all connected direct graph bundles ▫$X=C_stimes^{alpha}C_t$▫ are Hamiltonian