The Borodin-Kostochka conjecture for graphs containing a doubly critical edge.
The chromatic equivalence class of graph
By h(G,x) and P(G,λ) we denote the adjoint polynomial and the chromatic polynomial of graph G, respectively. A new invariant of graph G, which is the fourth character R₄(G), is given in this paper. Using the properties of the adjoint polynomials, the adjoint equivalence class of graph is determined, which can be regarded as the continuance of the paper written by Wang et al. [J. Wang, R. Liu, C. Ye and Q. Huang, A complete solution to the chromatic equivalence class of graph , Discrete Math....
The Chromatic Number of a Class of Pseudo-2-Manifolds.
The chromatic number of extended odd graphs is four
The Chromatic Number of Random Intersection Graphs
We study problems related to the chromatic number of a random intersection graph G (n,m, p). We introduce two new algorithms which colour G (n,m, p) with almost optimum number of colours with probability tending to 1 as n → ∞. Moreover we find a range of parameters for which the chromatic number of G (n,m, p) asymptotically equals its clique number.
The chromatic number of the product of two graphs is at least half the minimum of the fractional chromatic numbers of the factors
One consequence of Hedetniemi’s conjecture on the chromatic number of the product of graphs is that the bound should always hold. We prove that .
The chromatic sum of a graph: history and recent developments.
The chromaticity of a family of 2-connected 3-chromatic graphs with five triangles and cyclomatic number six
In this note, all chromatic equivalence classes for 2-connected 3-chromatic graphs with five triangles and cyclomatic number six are described. New families of chromatically unique graphs of order n are presented for each n ≥ 8. This is a generalization of a result stated in [5]. Moreover, a proof for the conjecture posed in [5] is given.
The circular chromatic index of flower snarks.
The color-balanced spanning tree problem
Suppose a graph whose edges are partitioned into disjoint categories (colors) is given. In the color-balanced spanning tree problem a spanning tree is looked for that minimizes the variability in the number of edges from different categories. We show that polynomiality of this problem depends on the number of categories and present some polynomial algorithm.
The cost chromatic number and hypergraph parameters
In a graph, by definition, the weight of a (proper) coloring with positive integers is the sum of the colors. The chromatic sum is the minimum weight, taken over all the proper colorings. The minimum number of colors in a coloring of minimum weight is the cost chromatic number or strength of the graph. We derive general upper bounds for the strength, in terms of a new parameter of representations by edge intersections of hypergraphs.
The cover-index of infinite graphs.
The directed path partition conjecture
The Directed Path Partition Conjecture is the following: If D is a digraph that contains no path with more than λ vertices then, for every pair (a,b) of positive integers with λ = a+b, there exists a vertex partition (A,B) of D such that no path in D⟨A⟩ has more than a vertices and no path in D⟨B⟩ has more than b vertices. We develop methods for finding the desired partitions for various classes of digraphs.
The distinguishing chromatic number.
The excessive [3]-index of all graphs.
The face labeling of maximal planar graphs.
The Fan-Raspaud conjecture: A randomized algorithmic approach and application to the pair assignment problem in cubic networks
It was conjectured by Fan and Raspaud (1994) that every bridgeless cubic graph contains three perfect matchings such that every edge belongs to at most two of them. We show a randomized algorithmic way of finding Fan-Raspaud colorings of a given cubic graph and, analyzing the computer results, we try to find and describe the Fan-Raspaud colorings for some selected classes of cubic graphs. The presented algorithms can then be applied to the pair assignment problem in cubic computer networks. Another...
The graph polynomial and the number of proper vertex coloring
Alon and Tarsi presented in a previous paper a certain weighted sum over the set of all proper -colorings of a graph, which can be computed from its graph polynomial. The subject of this paper is another combinatorial interpretation of the same quantity, expressed in terms of the numbers of certain modulo flows in the graph. Some relations between graph parameters can be obtained by combining these two formulas. For example: The number of proper 3-colorings of a 4-regular graph and the number...
The Grötzsch theorem for the hypergraph of maximal cliques.
The Hadwiger number of an infinite graph