Strong edge-coloring of cubic bipartite graphs: A counterexample

Strong edge-coloring for cubic Halin graphs

A strong edge-coloring of a graph G is a function that assigns to each edge a color such that two edges within distance two apart must receive different colors. The minimum number of colors used in a strong edge-coloring is the strong chromatic index of G. Lih and Liu [14] proved that the strong chromatic index of a cubic Halin graph, other than two special graphs, is 6 or 7. It remains an open...

Edge-Coloring Bipartite Graphs

Given a bipartite graph G with n nodes, m edges and maximum degree ∆, we find an edge coloring for G using ∆ colors in time T +O(m log ∆), where T is the time needed to find a perfect matching in a k-regular bipartite graph with O(m) edges and k ≤ ∆. Together with best known bounds for T this implies an O(m log ∆ + m ∆ log m ∆ log ∆) edge-coloring algorithm which improves on the O(m log ∆+ m ∆ ...

Strong edge-coloring of $(3, \Delta)$-bipartite graphs

A strong edge-coloring of a graph G is an assignment of colors to edges such that every color class induces a matching. We here focus on bipartite graphs whose one part is of maximum degree at most 3 and the other part is of maximum degree ∆. For every such graph, we prove that a strong 4∆-edge-coloring can always be obtained. Together with a result of Steger and Yu, this result confirms a conj...

Strong edge-coloring of (3, Δ)-bipartite graphs

Edge Coloring Bipartite Graphs Eeciently

The chromatic index of a bipartite graph equals the maximal degree of its vertices. The straightforward way to compute the corresponding edge coloring using colors, requires O((2 n 3=2) time. We will show that a simple divide & conquer algorithm only requires O((3=2 n 3=2) time. This algorithm uses an algorithm for perfect k-matching in regular bipartite graphs as a sub-routine. We will show th...