Shailesh K. Tipnis

The chromatic index of multigraphs of order at most 10

Abstract The maximum of the maximum degree and the “odd set quotients” provides a well-known lower bound φ ( G ) for the chromatic index of a multigraph G . Plantholt proved that if G is a multigraph of order at most 8, its chromatic index equals φ ( G ) and that if G is a multigraph of order 10, the chromatic index of G cannot exceed φ ( G ) + 1. We identify those multigraphs G of order 9 and 10 whose chromatic index equals φ ( G ) + 1, thus completing the determination of the chromatic index of all multigraphs of order at most 10.

On Decomposing Regular Graphs Into Isomorphic Double-Stars

Abstract A double-star is a tree with exactly two vertices of degree greater than 1. If T is a double-star where the two vertices of degree greater than one have degrees k1+1 and k2+1, then T is denoted by Sk1,k2 . In this note, we show that every double-star with n edges decomposes every 2n-regular graph. We also show that the double-star Sk,k−1 decomposes every 2k-regular graph that contains a perfect matching.

Improved Sufficient Conditions for the Existence of Anti-Directed Hamiltonian Cycles in Digraphs

Let D be a directed graph of order n. An anti-directed (hamiltonian) cycle H in D is a (hamiltonian) cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. In this paper we give sufficient conditions for the existence of anti-directed hamiltonian cycles. Specifically, we prove that a directed graph D of even order n with minimum indegree and outdegree greater than

On decomposing even regular multigraphs into small isomorphic trees

{\frac{1}{2}n + 7\sqrt{n}/3}

The chromatic index of multigraphs that are nearly full

contains an anti-directed hamiltonian cycle. In addition, we show that D contains anti-directed cycles of all possible (even) lengths when n is sufficiently large and has minimum in- and out-degree at least

A generalization of the Oberwolfach problem

{(1/2+ \epsilon)n}