Hadwiger's Conjecture for ℓ‐Link Graphs

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 ℓ‐link of G is a walk of G of length ℓ⩾0 in which consecutive edges are different. The ℓ‐link graph double-struckLℓfalse(Gfalse) of G is the graph with vertices the ℓ‐links of G, such that two vertices are joined by μ⩾0 edges in double-struckLℓfalse(Gfalse) if they correspond to two subsequences of each of μ (ℓ+1)‐links of G. By revealing a recursive structure, we bound from above the chromatic number of ℓ‐link graphs. As a corollary, for a given graph G and large enough ℓ, double-struckLℓfalse(Gfalse) is 3‐colorable. By investigating the shunting of ℓ‐links in G, we show that the Hadwiger number of a nonempty double-struckLℓfalse(Gfalse) 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 ℓ‐link graphs.

A study of link graphs

“…As a generalisation of line graphs [49] and path graphs [7], the ℓ-link graph of a given graph was introduced by Jia and Wood [23] who studied the connectedness, chromatic number and minors of ℓ-link graphs based on the structure of the given graph. This paper deals with the reverse; that is, for an integer ℓ 0 and a given finite graph H, we study the graphs whose ℓ-link graphs are isomorphic to H. Unless stated otherwise, all graphs are undirected and loopless.…”

“…Introduced by Broersma and Hoede [7], the ℓ-path graph P ℓ (G) is the simple graph with vertices the ℓ-paths of G, where two vertices are adjacent if the union of their corresponding paths forms a path or a cycle of length ℓ + 1 in G. By Jia and Wood [23], when ℓ 2, we have P ℓ (G) L ℓ (G), where the equation holds if and only if girth(G) > ℓ. We say G is an ℓ-path root of H if P ℓ (G) H. Let Q ℓ (H) be the set of minimal (up to subgraph relation) ℓ-path roots of H. Li [30] proved that H has at most one simple 2-path root of minimum degree at least 3.…”

“…Two general questions associated with a given incidence pattern were proposed by Grünbaum as characterising all graphs that can be constructed from this pattern, and determining the original object for each of these graphs. This thesis contributes in various aspects to the two questions for a certain incidence pattern, called the ℓ-link graph construction [23].…”

“…A sufficient and necessary condition for an ℓ-link graph to be connected was given by Jia and Wood [23]. In this section, we study the cyclic components of ℓ-roots.…”

“…In particular, an ℓ-path is an ℓ-link without repeated vertices. The ℓ-link graph L ℓ (G) of a graph G is defined to have vertices the ℓ-links of G, and two vertices are adjacent if their corresponding ℓ-links form an (ℓ + 1)-link of G. If further G contains parallel edges, then two ℓ-links may form µ 2 different (ℓ + 1)-links[23]. In this case, we give µ edges between the two corresponding vertices in L ℓ(G).…”