The concept of the incidence chromatic number of a graph was introduced by Brualdi and Massey (1993). They conjectured that every graph G can be incidence colored with ∆(G) + 2 colors. In this paper, we calculate the incidence chromatic numbers of the complete k-partite graphs and give the incidence chromatic number of three infinite families of graphs.

We investigate the cop number of graphs based on combinatorial designs. Incidence graphs, point graphs, and block intersection graphs are studied, with an emphasis on finding families of graphs with large cop number. We generalize known results on so-called Meyniel extremal families by supplying bounds on the incidence graph of transversal designs, certain G-designs, and BIBDs with λ ≥ 1. Famil...

The nnnrlnip of inclusion and exclusion and Consecutive Sets M. H. McIlwain )p'n,Clrj·'Yli'·nl of Mathematics Furman University Greenville, SC USA 29613 applied to numerous areas of discrete and combinatorial mathematics. One manifestation of this occurs in probability of the union of events an alternating sum of pn)o(HHjlIW~S of various intersections of events. If the constituent events are th...

We prove that the number of incidences between m points and n bounded-degree curves with k degrees of freedom in R is

Let P be a set of m points and L a set of n lines in R, such that the points of P lieon an algebraic three-dimensional surface of degree D that does not contain hyperplaneor quadric components, and no 2-flat contains more than s lines of L. We show thatthe number of incidences between P and L is

In this article, we prove that amongst all n by n bipartite graphs of girth at least six, where n = q + q + 1 ≥ 157, the incidence graph of a projective plane of order q, when it exists, has the maximum number of cycles of length eight. This characterizes projective planes as the partial planes with the maximum number of quadrilaterals.

The questions of isomorphism degree estimation of fuzzy graphs are discussed in this paper. The definition of independent fuzzy set of fuzzy graph is presented. The estimation method of isomorphism degree of fuzzy graphs is suggested and proved. The example of isomorphism degree estimation is considered.

We prove two theorems concerning incidence posets of graphs, cover graphs of posets and a related graph parameter. First, answering a question of Haxell, we show that the chromatic number of a graph is not bounded in terms of the dimension of its incidence poset, provided the dimension is at least four. Second, answering a question of Kříž and Nešetřil, we show that there are graphs with large ...

In a graph Γ = (V,E) a vertex v is resolved by a vertex-set S = {v1, . . . , vn} if its (ordered) distance list with respect to S, (d(v, v1), . . . , d(v, vn)), is unique. A set A ⊂ V is resolved by S if all its elements are resolved by S. S is a resolving set in Γ if it resolves V . The metric dimension of Γ is the size of the smallest resolving set in it. In a bipartite graph a semi-resolving...

Two inequalities are established connecting the graph invariants of incidence chromatic number, star arboricity and domination number. Using these, upper and lower bounds are deduced for the incidence chromatic number of a graph and further reductions are made to the upper bound for a planar graph. It is shown that cubic graphs with orders not divisible by four are not 4-incidence colorable. Sh...