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Hardness Results for Total Rainbow Connection of Graphs

Lily Chen, Bofeng Huo, Yingbin Ma (2016)

Discussiones Mathematicae Graph Theory

A total-colored path is total rainbow if both its edges and internal vertices have distinct colors. The total rainbow connection number of a connected graph G, denoted by trc(G), is the smallest number of colors that are needed in a total-coloring of G in order to make G total rainbow connected, that is, any two vertices of G are connected by a total rainbow path. In this paper, we study the computational complexity of total rainbow connection of graphs. We show that deciding whether a given total-coloring...

Improved Sufficient Conditions for Hamiltonian Properties

Jens-P. Bode, Anika Fricke, Arnfried Kemnitz (2015)

Discussiones Mathematicae Graph Theory

In 1980 Bondy [2] proved that a (k+s)-connected graph of order n ≥ 3 is traceable (s = −1) or Hamiltonian (s = 0) or Hamiltonian-connected (s = 1) if the degree sum of every set of k+1 pairwise nonadjacent vertices is at least ((k+1)(n+s−1)+1)/2. It is shown in [1] that one can allow exceptional (k+ 1)-sets violating this condition and still implying the considered Hamiltonian property. In this note we generalize this result for s = −1 and s = 0 and graphs that fulfill a certain connectivity condition....

Interpolation theorem for a continuous function on orientations of a simple graph

Fu Ji Zhang, Zhibo Chen (1998)

Czechoslovak Mathematical Journal

Let G be a simple graph. A function f from the set of orientations of G to the set of non-negative integers is called a continuous function on orientations of G if, for any two orientations O 1 and O 2 of G , | f ( O 1 ) - f ( O 2 ) | ≤ 1 whenever O 1 and O 2 differ in the orientation of exactly one edge of G . We show that any continuous function on orientations of a simple graph G has the interpolation property as follows: If there are two orientations O 1 and O 2 of G with f ( O 1 ) = p and f ( O 2 ) = q , where p < q , then for any integer k such that p < k < q , there are...

Intersection graph of gamma sets in the total graph

T. Tamizh Chelvam, T. Asir (2012)

Discussiones Mathematicae Graph Theory

In this paper, we consider the intersection graph I Γ ( ℤ ₙ ) of gamma sets in the total graph on ℤₙ. We characterize the values of n for which I Γ ( ℤ ₙ ) is complete, bipartite, cycle, chordal and planar. Further, we prove that I Γ ( ℤ ₙ ) is an Eulerian, Hamiltonian and as well as a pancyclic graph. Also we obtain the value of the independent number, the clique number, the chromatic number, the connectivity and some domination parameters of I Γ ( ℤ ₙ ) .

Isometric classification of Sobolev spaces on graphs

M. I. Ostrovskii (2007)

Colloquium Mathematicae

Isometric Sobolev spaces on finite graphs are characterized. The characterization implies that the following analogue of the Banach-Stone theorem is valid: if two Sobolev spaces on 3-connected graphs, with the exponent which is not an even integer, are isometric, then the corresponding graphs are isomorphic. As a corollary it is shown that for each finite group and each p which is not an even integer, there exists n ∈ ℕ and a subspace L ⊂ ℓ ⁿ p whose group of isometries is the direct product × ℤ₂.

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