Characterizing symmetric diametrical graphs of order 12 and diameter 4.
Given a graph G = (V,E) and a set Lv of admissible colors for each vertex v ∈ V (termed the list at v), a list coloring of G is a (proper) vertex coloring ϕ : V → S v2V Lv such that ϕ(v) ∈ Lv for all v ∈ V and ϕ(u) 6= ϕ(v) for all uv ∈ E. If such a ϕ exists, G is said to be list colorable. The choice number of G is the smallest natural number k for which G is list colorable whenever each list contains at least k colors. In this note we initiate the study of graphs in which the choice number equals...
A graph is ptolemaic if and only if it is both chordal and distance-hereditary. Thus, a ptolemaic graph G has two kinds of intersection graph representations: one from being chordal, and the other from being distance-hereditary. The first of these, called a clique tree representation, is easily generated from the clique graph of G (the intersection graph of the maximal complete subgraphs of G). The second intersection graph representation can also be generated from the clique graph, as a very special...
In this paper we study the chromatic number of graphs with two prescribed induced cycle lengths. It is due to Sumner that triangle-free and P₅-free or triangle-free, P₆-free and C₆-free graphs are 3-colourable. A canonical extension of these graph classes is , the class of all graphs whose induced cycle lengths are 4 or 5. Our main result states that all graphs of are 3-colourable. Moreover, we present polynomial time algorithms to 3-colour all triangle-free graphs G of this kind, i.e., we have...
The -core of a graph , , is the maximal induced subgraph such that , if it exists. For , the -shell of a graph is the subgraph of induced by the edges contained in the -core and not contained in the -core. The core number of a vertex is the largest value for such that , and the maximum core number of a graph, , is the maximum of the core numbers of the vertices of . A graph is -monocore if . This paper discusses some basic results on the structure of -cores and -shells....
A graph is called splitting if there is a 0-1 labelling of its vertices such that for every infinite set C of natural numbers there is a sequence of labels along a 1-way infinite path in the graph whose restriction to C is not eventually constant. We characterize the countable splitting graphs as those containing a subgraph of one of three simple types.
Let ₁,₂,...,ₙ be graph properties, a graph G is said to be uniquely (₁,₂, ...,ₙ)-partitionable if there is exactly one (unordered) partition V₁,V₂,...,Vₙ of V(G) such that for i = 1,2,...,n. We prove that for additive and induced-hereditary properties uniquely (₁,₂,...,ₙ)-partitionable graphs exist if and only if and are either coprime or equal irreducible properties of graphs for every i ≠ j, i,j ∈ 1,2,...,n.
Let V be a finite vertex set and let (, +) be a finite abelian group. An -labeled and reversible 2-structure defined on V is a function g : (V × V) (v, v) : v ∈ V → such that for distinct u, v ∈ V, g(u, v) = −g(v, u). The set of -labeled and reversible 2-structures defined on V is denoted by ℒ(V, ). Given g ∈ ℒ(V, ), a subset X of V is a clan of g if for any x, y ∈ X and v ∈ V X, g(x, v) = g(y, v). For example, ∅, V and v (for v ∈ V) are clans of g, called trivial. An element g of ℒ(V, ) is primitive...
Let G = (V,A) be a directed graph. With any subset X of V is associated the directed subgraph G[X] = (X,A ∩ (X×X)) of G induced by X. A subset X of V is an interval of G provided that for a,b ∈ X and x ∈ V∖X, (a,x) ∈ A if and only if (b,x) ∈ A, and similarly for (x,a) and (x,b). For example ∅, V, and {x}, where x ∈ V, are intervals of G which are the trivial intervals. A directed graph is indecomposable if all its intervals are trivial. Given an integer k > 0, a directed graph G = (V,A) is called...
A k-monocore graph is a graph which has its minimum degree and degeneracy both equal to k. Integer sequences that can be the degree sequence of some k-monocore graph are characterized as follows. A nonincreasing sequence of integers d0, . . . , dn is the degree sequence of some k-monocore graph G, 0 ≤ k ≤ n − 1, if and only if k ≤ di ≤ min {n − 1, k + n − i} and ⨊di = 2m, where m satisfies [...] ≤ m ≤ k ・ n − [...] .