For a graph G, let M(G) denote the Mycielski graph of G. The t-th iterated Mycielski graph of G, M(G), is defined recursively by M0(G) = G and M(G)= M(Mt−1(G)) for t ≥ 1. Let χc(G) denote the circular chromatic number of G. We prove two main results: 1) Assume G has a universal vertex x, then χc(M(G)) = χ(M(G)) if χc(G − x) > χ(G − x) − 1/2 and G is not a star, otherwise χc(M(G)) = χ(M(G)) − 1/...

Mycielski introduced a new graph transformation μ(G) for graph G, which is called the Mycielskian of G. A graph G is super connected or simply super-κ (resp. super edge connected or super-λ), if every minimum vertex cut (resp. minimum edge cut) isolates a vertex of G. In this paper, we show that for a connected graph G with |V (G)| ≥ 2, μ(G) is super-κ if and only if δ(G) < 2κ(G), and μ(G) is s...

We give a simple, direct, and constructive proof of memoryless determinacy for parity and mean payo& games. First, we prove by induction that the 8nite duration versions of these games, played until some vertex is repeated, are determined and both players have memoryless winning strategies. In contrast to the proof of Ehrenfeucht and Mycielski, Internat. J. Game Theory, 8 (1979) 109–113, our pr...

Zykov designed one of the oldest known family of triangle-free graphs with arbitrarily high chromatic number. We determine the fractional chromatic number of the Zykov product of a family of graphs. As a corollary, we deduce that the fractional chromatic numbers of the Zykov graphs satisfy the same recurrence relation as those of the Mycielski graphs, that is an+1 = an + 1 an . This solves a co...

We introduce the inverted prefix tries (a variation of suffix tries) as a convenient formalism for stating and proving properties of the EhrenfeuchtMycielski sequence ([3]). We also prove an upper bound on the position in the sequence by which all strings of a given length will have appeared; our bound is given by the Ackermann function, which, in light of experimental data, may be a gross over...

The technique of counting cliques in networks is a natural problem. In this paper, we develop certain results on counting of triangles for the total graph of the Mycielski graph or central graph of star as well as completegraph families. Moreover, we discuss the upper bounds for the number of triangles in the Mycielski and other well known transformations of graphs. Finally, it is shown that th...

It is shown that the first-order theory T of a non-surjective injection (of the universe into itself) is connected. By a criterion for connectedness, due to J. Mycielski, one reduces this result to the following theorem: If A, B are any structures and ε is a definable equivalence relation on their disjoint product A ×̇B such that (A ×̇B)/ε is a model of T, then either A has the property that for ...