The weak critical graph conjecture [1,7] claims that there exists a constant c > 0 such that every critical multigraph M with at most c • ∆(M ) vertices has odd order. We disprove this conjecture by constructing critical multigraphs of order 20 with maximum degree k for all k ≥ 5.
Chromatic-index-critical graphs of even order
✍ Scribed by Gr�newald, Stefan; Steffen, Eckhard
- Publisher
- John Wiley and Sons
- Year
- 1999
- Tongue
- English
- Weight
- 298 KB
- Volume
- 30
- Category
- Article
- ISSN
- 0364-9024
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✦ Synopsis
A k-critical (multi-) graph G has maximum degree k, chromatic index χ (G) = k + 1, and χ (G -e) < k + 1 for each edge e of G. For each k ≥ 3, we construct k-critical (multi-) graphs with certain properties to obtain counterexamples to some well-known conjectures.
📜 SIMILAR VOLUMES
The strong chromatic index of a graph G, denoted sq(G), is the minimum number of parts needed to partition the edges of G into induced matchings. For 0 ≤ k ≤ l ≤ m, the subset graph S m (k, l) is a bipartite graph whose vertices are the kand l-subsets of an m element ground set where two vertices ar
We show that coloring the edges of a multigraph G in a particular order often leads to improved upper bounds for the chromatic index χ (G). Applying this to simple graphs, we significantly generalize recent conditions based on the core of G (i.e., the subgraph of G induced by the vertices of degree