Abstract. Let Γ be a finite X-symmetric graph with a nontrivial Xinvariant partition B on V (Γ) such that ΓB is a connected (X, 2)-arctransitive graph and Γ is not a multicover of ΓB. A characterization of (Γ, X, B) was given in [20] for the case where |Γ(C) ∩ B| = 2 for B ∈ B and C ∈ ΓB(B). This motivates us to investigate the case where |Γ(C) ∩ B| = 3, that is, Γ[B, C] is isomorphic to one of 3K2, K3,3 − 3K2 and K3,3. This investigation requires a study on (X, 2)-arc-transitive graphs of valency 4 or 7. Based on the results in [14], we give a characterization of tetravalent (X, 2)-arc-transitive graphs; and as a byproduct, we prove that every tetravalent (X, 2)-transitive graph is either the complete graph on 5 vertices or a near n-gonal graph for some n ≥ 4. We show that a heptavalent (X, 2)-arc-transitive graph Σ can occur as ΓB if and only if X Σ(τ ) τ ∼ = P SL(3, 2) for τ ∈ V (Σ).
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.
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