A class of symmetric graphs with 2‐arc transitive quotients

Jia, Bin; Lu, Zai Ping; Wang, Gai Xia

doi:10.1002/jgt.20476

Cited by 8 publications

(9 citation statements)

References 21 publications

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“…The

ℓ t h

iterated line digraph

{double-struckA}^{ℓ} false(G false)

{double-struckA}_{1} false(G false)

ℓ = 1

, and

{double-struckA}_{1} false(A^{ℓ - 1} (G) false)

ℓ ⩾ 2

(see ). Examples of undirected graphs constructed from ℓ‐arcs can be found in .…”

Section: Definitions and Terminologymentioning

confidence: 99%

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Hadwiger's Conjecture for ℓ‐Link Graphs

Jia

Wood

2016

Journal of Graph Theory

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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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“…The

ℓ t h

iterated line digraph

{double-struckA}^{ℓ} false(G false)

{double-struckA}_{1} false(G false)

ℓ = 1

, and

{double-struckA}_{1} false(A^{ℓ - 1} (G) false)

ℓ ⩾ 2

(see ). Examples of undirected graphs constructed from ℓ‐arcs can be found in .…”

Section: Definitions and Terminologymentioning

confidence: 99%

“…(See Figure 1 a for D 3 , and Figure 1 [2]). Examples of undirected graphs constructed from -arcs can be found in [12,13].…”

Section: Definitions and Terminologymentioning

confidence: 99%

Hadwiger's Conjecture for ℓ‐Link Graphs

Jia

Wood

2016

Journal of Graph Theory

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“…In the cases of M 11 and M 12 , a is 2, 3, 4 or 6, and so (v, b, r, λ, r * ) = (22, 12, 6, 5, 11), (33,12,8,14,22), (44,12,9,24,33) This gives the last row in Table 2.…”

Section: Introductionmentioning

confidence: 99%

“…or (66,12,10, 45, 55). Since by [4, Section II.1.3] a 2-(12,8,14) or 2-(12,9, 24) design does not exist, the second and third possibilities can be eliminated. Thus, if G Γ B (B) B ∼ = M 11 or M 12 , then D * (B) is isomorphic to a 2-(12, 6, 5) or 2-(12, 10, 45) design.…”

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confidence: 99%

Symmetric Graphs With 2-Arc Transitive Quotients

Zhou

2014

J. Aust. Math. Soc.

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A graph Γ is G-symmetric if Γ admits G as a group of automorphisms acting transitively on the set of vertices and the set of arcs of Γ, where an arc is an ordered pair of adjacent vertices. In the case when G is imprimitive on V (Γ), namely when V (Γ) admits a nontrivial G-invariant partition B, the quotient graph Γ B of Γ with respect to B is always G-symmetric and sometimes even (G, 2)-arc transitive. (A G-symmetric graph is (G, 2)-arc transitive if G is transitive on the set of oriented paths of length two.) In this paper we obtain necessary conditions for Γ B to be (G, 2)-arc transitive (regardless of whether Γ is (G, 2)-arc transitive) in the case when v − k is an odd prime p, where v is the block size of B and k is the number of vertices in a block having neighbours in a fixed adjacent block. These conditions are given in terms of v, k and two other parameters with respect to (Γ, B) together with a certain 2-point transitive block design induced by (Γ, B). We prove further that if p = 3 or 5 then these necessary conditions are essentially sufficient for Γ B to be (G, 2)-arc transitive.

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“…In particular, if r = 1, then Γ is 2-arc-transitive. The first remarkable result about 2-arc-transitive graphs comes from Tutte [19,20], and this family of graphs has been studied extensively, see [1,9,10,12,[15][16][17][18]21]. The graphs in case (1) were investigated in [14]; and in [13], the author completely determined such graphs with valency twice a prime.…”

Section: Introductionmentioning

confidence: 99%