A radio labeling of a graph G is a mapping f : V ( G ) ź { 0 , 1 , 2 , ź } such that | f ( u ) - ... more A radio labeling of a graph G is a mapping f : V ( G ) ź { 0 , 1 , 2 , ź } such that | f ( u ) - f ( v ) | ź diam ( G ) + 1 - d ( u , v ) for every pair of distinct vertices u , v of G , where diam ( G ) is the diameter of G and d ( u , v ) the distance between u and v in G . The radio number of G is the smallest integer k such that G has a radio labeling f with max { f ( v ) : v ź V ( G ) } = k . We give a necessary and sufficient condition for a lower bound on the radio number of trees to be achieved, two other sufficient conditions for the same bound to be achieved by a tree, and an upper bound on the radio number of trees. Using these, we determine the radio number for three families of trees.
A graph Γ is called (G, s)-arc-transitive if G ≤Aut(Γ) is transitive on the set of vertices of Γ ... more A graph Γ is called (G, s)-arc-transitive if G ≤Aut(Γ) is transitive on the set of vertices of Γ and the set of s-arcs of Γ, where for an integer s ≥ 1 an s-arc of Γ is a sequence of s+1 vertices (v_0,v_1,…,v_s) of Γ such that v_i-1 and v_i are adjacent for 1 ≤ i ≤ s and v_i-1 v_i+1 for 1 ≤ i ≤ s-1. Γ is called 2-transitive if it is (Aut(Γ), 2)-arc-transitive but not (Aut(Γ), 3)-arc-transitive. A Cayley graph Γ of a group G is called normal if G is normal in Aut(Γ) and non-normal otherwise. It was proved by X. G. Fang, C. H. Li and M. Y. Xu that if Γ is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either Γ is normal or G is one of the groups PSL_2(11), M_11, M_23 and A_11. However, it was unknown whether Γ is normal when G is one of these four groups. In the present paper we answer this question by proving that among these four groups only M_11 produces connected tetravalent 2-transitive non-normal Cayley graphs. We prove further that there are exactly tw...
Hadwiger’s conjecture states that for every graph G, χ(G) ≤ η(G), where χ(G) is the chromatic num... more Hadwiger’s conjecture states that for every graph G, χ(G) ≤ η(G), where χ(G) is the chromatic number and η(G) is the size of the largest clique minor in G. In this work, we show that to prove Hadwiger’s conjecture in general, it is sufficient to prove Hadwiger’s conjecture for the class of graphs F defined as follows: F is the set of all graphs that can be expressed as the square graph of a chordal graph. Hence, it is interesting to study Hadwiger’s Conjecture in the square graphs of subclasses of chordal graphs. Here, we study a simple subclass of chordal graphs, namely 2-trees and prove Hadwiger’s Conjecture for the square of the same. In fact, we show the following stronger result: If G is the square of a 2-tree, then G has a clique minor of size χ(G), where each branch set is a path.
A radio labeling of a graph G is a mapping f : V ( G ) ź { 0 , 1 , 2 , ź } such that | f ( u ) - ... more A radio labeling of a graph G is a mapping f : V ( G ) ź { 0 , 1 , 2 , ź } such that | f ( u ) - f ( v ) | ź diam ( G ) + 1 - d ( u , v ) for every pair of distinct vertices u , v of G , where diam ( G ) is the diameter of G and d ( u , v ) the distance between u and v in G . The radio number of G is the smallest integer k such that G has a radio labeling f with max { f ( v ) : v ź V ( G ) } = k . We give a necessary and sufficient condition for a lower bound on the radio number of trees to be achieved, two other sufficient conditions for the same bound to be achieved by a tree, and an upper bound on the radio number of trees. Using these, we determine the radio number for three families of trees.
A graph Γ is called (G, s)-arc-transitive if G ≤Aut(Γ) is transitive on the set of vertices of Γ ... more A graph Γ is called (G, s)-arc-transitive if G ≤Aut(Γ) is transitive on the set of vertices of Γ and the set of s-arcs of Γ, where for an integer s ≥ 1 an s-arc of Γ is a sequence of s+1 vertices (v_0,v_1,…,v_s) of Γ such that v_i-1 and v_i are adjacent for 1 ≤ i ≤ s and v_i-1 v_i+1 for 1 ≤ i ≤ s-1. Γ is called 2-transitive if it is (Aut(Γ), 2)-arc-transitive but not (Aut(Γ), 3)-arc-transitive. A Cayley graph Γ of a group G is called normal if G is normal in Aut(Γ) and non-normal otherwise. It was proved by X. G. Fang, C. H. Li and M. Y. Xu that if Γ is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either Γ is normal or G is one of the groups PSL_2(11), M_11, M_23 and A_11. However, it was unknown whether Γ is normal when G is one of these four groups. In the present paper we answer this question by proving that among these four groups only M_11 produces connected tetravalent 2-transitive non-normal Cayley graphs. We prove further that there are exactly tw...
Hadwiger’s conjecture states that for every graph G, χ(G) ≤ η(G), where χ(G) is the chromatic num... more Hadwiger’s conjecture states that for every graph G, χ(G) ≤ η(G), where χ(G) is the chromatic number and η(G) is the size of the largest clique minor in G. In this work, we show that to prove Hadwiger’s conjecture in general, it is sufficient to prove Hadwiger’s conjecture for the class of graphs F defined as follows: F is the set of all graphs that can be expressed as the square graph of a chordal graph. Hence, it is interesting to study Hadwiger’s Conjecture in the square graphs of subclasses of chordal graphs. Here, we study a simple subclass of chordal graphs, namely 2-trees and prove Hadwiger’s Conjecture for the square of the same. In fact, we show the following stronger result: If G is the square of a 2-tree, then G has a clique minor of size χ(G), where each branch set is a path.
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