(a) Perhaps the most important topic in extremal graph theory, is to determine the Turán number of a graph H . This is the maximal number of edges a graph of size n can have, without having a subgraph isomorphic to H . When H is non-bipartite, the asymptotics of this function is determined by the celebrated Erdös-Stone theorem, up to an 1 + o(1) factor. However, it is a longstanding problem to ...

Borowiecki, M. and E. Drgas-Burchardt, Classes of chromatically unique graphs, Discrete Mathematics Ill (1993) 71-75. We prove that graphs obtained from complete equibipartite graphs by deleting some independent sets of edges are chromatically unique. 1. Preliminary definitions and results In this paper we consider finite, undirected, simple and loopless graphs. Two graphs G and H are said to b...

Let G be a 2-connected graph of order n. For any u ∈ V (G) and l ∈ {m,m + 1, . . . , n}, if G has a cycle of length l, then G is called [m,n]pancyclic, and if G has a cycle of length l which contains u, then G is called [m,n]-vertex pancyclic. Let δ(G) be a minimum degree ofG and let NG(x) be the neighborhood of a vertex x in G. In [Australas. J. Combin. 12 (1995), 81–91] Liu, Lou and Zhao prov...

Heterochromatic brightness matches with checkerboard patterns made up of a white reference and a chromatic test element for various element sizes from 3' to 2 degrees 30' square and with a 5 degrees bipartite field were carried out with a color cathode-ray tube. The brightness-to-luminance (B/L) ratios of six chromatic stimuli were obtained, and the brightness additivity for the mixture of red ...

let g=(v,e) be a simple graph. an edge labeling f:e to {0,1} induces a vertex labeling f^+:v to z_2 defined by $f^+(v)equiv sumlimits_{uvin e} f(uv)pmod{2}$ for each $v in v$, where z_2={0,1} is the additive group of order 2. for $iin{0,1}$, let e_f(i)=|f^{-1}(i)| and v_f(i)=|(f^+)^{-1}(i)|. a labeling f is called edge-friendly if $|e_f(1)-e_f(0)|le 1$. i_f(g)=v_f(1)-v_f(0) is called the edge-f...

Franzkeit, R., A binary search problem on graphs, Discrete Applied Mathematics 36 (1992) 83-86. Let G = (V, E) be a graph and e’~ E an unknown edge. I:, order to find e’ we choose a sequence of test sets WC V, where aftev every test we are told whether both vertices incident to e’ are in W, or not. For C(Gj, L,,~ Aminimum number of tests required, the inequality c(G)? rlogz 1~11 clearly holds (...

The Fibonacci cube Γn is a subgraph of n-dimensional hypercube induced by the vertices without two consecutive ones. Klavžar and Žigert [Fibonacci cubes are the resonance graphs of fibonaccenes, Fibonacci Quart. 43 (2005) 269–276] proved that Fibonacci cubes are precisely the Z -transformation graphs (or resonance graphs) of zigzag hexagonal chains. In this paper, we characterize plane bipartit...

We provide a new relaxation of conditions under which the inequality H(A|B,X) +H(A|B,Y ) 6 H(A|B) holds for jointly distributed random variables A,B,X, Y . Then we present two applications of our result. The first one is the following easy-to-formulate combinatorial theorem: Assume that edges of a bipartite graph are partitioned into K matchings so that for each pair (left vertex x, right verte...