A note on upper bound for chromatic number of a graph.
A note on upper embeddable graphs
A Note On Vertex Colorings Of Plane Graphs
Given an integer valued weighting of all elements of a 2-connected plane graph G with vertex set V , let c(v) denote the sum of the weight of v ∈ V and of the weights of all edges and all faces incident with v. This vertex coloring of G is proper provided that c(u) ≠ c(v) for any two adjacent vertices u and v of G. We show that for every 2-connected plane graph there is such a proper vertex coloring with weights in {1, 2, 3}. In a special case, the value 3 is improved to 2.
A note to independent sets in scheduling
The paper studies the bus-journey graphs in the case when they are piecewise expanding and contracting (if described by fathers-sons relations starting with the greatest independent set of nodes). This approach can make it possible to solve the minimization problem of the total service time of crews.
A numerical invariant for finitely generated groups via actions on graphs.
A -analogue of a formula of Frobenius.
A paintability version of the combinatorial Nullstellensatz, and list colorings of -partite -uniform hypergraphs.
A parallelogram configuration condition in nets
After describing a (general and special) coordinatization of -nets there are found algebraic equivalents for the validity of certain quadrangle configuration conditions in -nets with small degree .
A parametric analysis of the largest induced tree problem in random graphs
A parametrization of the abstract Ramsey theorem.
A partition identity with certain applications
A partition of connected graphs.
A partition of R in two homogeneous and homeomorphic parts
A partition of the Catalan numbers and enumeration of genealogical trees
A special relational structure, called genealogical tree, is introduced; its social interpretation and geometrical realizations are discussed. The numbers of all abstract genealogical trees with exactly n+1 nodes and k leaves is found by means of enumeration of code words. For each n, the form a partition of the n-th Catalan numer Cₙ, that means .
A partition of the non-negative integers, with applications.
A partition theorem for α-large sets
A path(ological) partition problem
Let τ(G) denote the number of vertices in a longest path of the graph G and let k₁ and k₂ be positive integers such that τ(G) = k₁ + k₂. The question at hand is whether the vertex set V(G) can be partitioned into two subsets V₁ and V₂ such that τ(G[V₁] ) ≤ k₁ and τ(G[V₂] ) ≤ k₂. We show that several classes of graphs have this partition property.
A permutation regularity lemma.
A permutations representation that knows what “Eulerian” means.
A Pfaffian-Hafnian analogue of Borchardt's identity.