A study of 3-arc graphs

Three-arc graphs: Characterization and domination

“…It was proved in [8] that, for any connected graph G of order n ≥ 4 and minimum degree at least 2, we have γ(X(G)) ≤ n. Combining this with n/(1 + ∆(G)) ≤ γ(G) [3], we then have γ(X(G)) ≤ (1 + ∆(G))γ(G).…”

“…With D j as above we now set D := ∪ s j=1 D j . As in (8), choose a set of arcs of G by setting A(D j ) = {xx : x ∈ D j }, where x is a neighbour of x in S. (As seen in the previous paragraph, such a set A(D j ) is unique when R j is isomorphic to some graph in A and V (R j ) ⊆ W .) Set A(D) = ∪ s j=1 A(D j ).…”

Hadwiger's Conjecture for 3-Arc Graphs

“…It was used in classifying or characterizing certain families of arc-transitive graphs [9,11,15,17,23,24,25]. Recently, various graph-theoretic properties of 3-arc graphs have been investigated [1,12,13,22].…”

“…The 3-arc graph construction can be generalised for a digraph D = (V (D), A(D)) as follows [12], where A(D) is a multiset of ordered pairs (namely, arcs) of distinct vertices of V (D). Here a digraph allows parallel arcs but not loops.…”

Hamiltonicity of 3-Arc Graphs

“…On the other hand, the quotient graph of X(G) with respect to the partition P = {{uv, vu} : {u, v} ∈ E(G)} of A(G) is isomorphic to the graph obtained from the square of L(G) by deleting the edges of L(G). The reader is referred to [14,13,2] respectively for results on the diameter and connectivity, the independence, domination and chromatic numbers, and the edge-connectivity and restricted edge-connectivity of 3-arc graphs.…”