Hadwiger's Conjecture for ℓ‐Link Graphs |
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Authors: | Bin Jia David R. Wood |
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Affiliation: | 1. DEPARTMENT OF MATHEMATICS AND STATISTICS, THE UNIVERSITY OF MELBOURNE, MELBOURNE, AUSTRALIA;2. SCHOOL OF MATHEMATICAL SCIENCES, MONASH UNIVERSITY, MELBOURNE, AUSTRALIA |
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Abstract: | In this article, we define and study a new family of graphs that generalizes the notions of line graphs and path graphs. Let G be a graph with no loops but possibly with parallel edges. An ?‐link of G is a walk of G of length in which consecutive edges are different. The ?‐link graph of G is the graph with vertices the ?‐links of G , such that two vertices are joined by edges in if they correspond to two subsequences of each of μ ‐links of G . By revealing a recursive structure, we bound from above the chromatic number of ?‐link graphs. As a corollary, for a given graph G and large enough ?, is 3‐colorable. By investigating the shunting of ?‐links in G , we show that the Hadwiger number of a nonempty is greater or equal to that of G . Hadwiger's conjecture states that the Hadwiger number of a graph is at least the chromatic number of that graph. The conjecture has been proved by Reed and Seymour (Eur J Combin 25(6) (2004), 873–876) for line graphs, and hence 1‐link graphs. We prove the conjecture for a wide class of ?‐link graphs. |
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Keywords: | ℓ ‐link graph path graph chromatic number graph minor Hadwiger's conjecture |
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