Let G be an edge-colored graph. A path P of G is said to be rainbow if no two edges of P have the same color. An edge-coloring of G is a rainbow t-coloring if for any two distinct vertices u and v of G there are at least t internally vertex-disjoint rainbow (u, v)-paths. The rainbow t-connectivity rc t (G) of a graph G is the minimum integer j such that there exists a rainbow t-coloring using j colors. A (k; g)-cage is a k-regular graph of girth g and minimum number of vertices denoted n(k; g). In this paper we focus on g = 6. It is known that n(k; 6) ≥ 2(k 2 − k + 1) and when n(k; 6) = 2(k 2 − k + 1) the (k; 6)-cage is called a Moore cage. In this paper we prove that the rainbow k-connectivity of a Moore (k; 6)-cage G satisfies that k ≤ rc k (G) ≤ k 2 − k + 1. It is also proved that the rainbow 3-connectivity of the Heawood graph is 6 or 7.
The complete twisted graph of order n, denoted by T n , is a complete simple topological graph with vertices u 1 ; u 2 ;. . .; u n such that two edges u i u j and u i 0 u j 0 cross if and only if i\i 0 \j 0 \j or i 0 \i\j\j 0. The convex geometric complete graph of order n, denoted by G n , is a convex geometric graph with vertices v 1 ; v 2 ;. . .; v n placed counterclockwise, in which every pair of vertices is adjacent. A biplanar tree of order n is a labeled tree with vertex set fv 1 ; v 2 ;. . .; v n g having the property of being planar when embedded in both T n and G n. Given a connected graph G the (combinatorial) tree graph T ðGÞ is the graph whose vertices are the spanning trees of G and two trees P and Q are adjacent in T ðGÞ if there are edges e 2 P and f 2 Q such that Q ¼ P À e þ f. For all positive integers n, we denote by T ðnÞ the graph T ðK n Þ. The biplanar tree graph, BðnÞ, is the subgraph of T ðnÞ induced by the biplanar trees of order n. In this paper we give a characterization of the biplanar trees and we study the structure, the radius and the diameter of the biplanar tree graph.
A 2-switch on a simple graph G consists of deleting two edges {u, v} and {x, y} of G and adding the edges {u, x} and {v, y}, provided the resulting graph is a simple graph. It is well known that if two graphs G and H have the same set of vertices and the same degree sequence, then H can be obtained from G by a finite sequence of 2-switches. While the 2-switch transformation preserves the degree sequence other conditions like connectivity may be lost. We study the restricted case where 2-switches are applied to trees to obtain trees.
The packing chromatic number χ ρ (G) of a graph G is the smallest integer k for which there exists a vertex coloring Γ : V (G) → {1, 2, . . . , k} such that any two vertices of color i are at distance at least i + 1. For g ∈ {6, 8, 12}, (q + 1, g)-Moore graphs are (q + 1)-regular graphs with girth g which are the incidence graphs of a symmetric generalized g/2-gons of order q. In this paper we study the packing chromatic number of a (q + 1, g)-Moore graph G. For g = 6 we present the exact value of χ ρ (G). For g = 8, we determine χ ρ (G) in terms of the intersection of certain structures in generalized quadrangles. For g = 12, we present lower and upper bounds for this invariant when q ≥ 9 an odd prime power.
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