On set colorings of complete bipartite graphs

In European J. Combin. 30 (2009), 986–995, S.M. Hegde recently introduced set colorings of a graph G as an assignment (function) of distinct subsets of a finite set X of colors to the vertices of G, where the colors of the edges are obtained as the symmetric difference of the sets assigned to their end vertices (which are also distinct). A set coloring is called a proper set coloring if all the nonempty subsets of X are obtained on the edges. A graph is called properly set colorable if it admits a proper set coloring. In this paper we give a proof for Hegde’s conjecture that the complete bipartite graph Ka,b is properly set colorable if and only if one of the partition sets is of cardinality 1, and the other one of cardinality 2 − 1 for some positive integer n. 1 Terminology and introduction In this paper we consider finite and simple graphs only. The vertex set and edge set of a graph G are denoted by V (G) and E(G), respectively. Let X be a nonempty ∗ Corresponding author † Supported by “Undergraduate Funds” of RWTH Aachen University, Germany. 246 S. GRÜTER, A. HOLTKAMP AND M. SURMACS set of colors and 2 denote the power set of X. For any two subsets Y, Z ⊆ X let Y ⊕ Z = (Y ∪ Z) \ (Y ∩ Z) denote the symmetric difference of Y and Z. Given a function f : V (G) → 2 we define f⊕ : E(G) → 2 by f⊕(uv) = f(u)⊕ f(v) for all edges uv ∈ E(G). We call f a set coloring of G if both f and f⊕ are injective functions. A graph is called set colorable if it admits a set coloring. A set coloring f is called a proper set coloring if f⊕(E(G)) = {f⊕(e)|e ∈ E(G)} = 2X\{∅}. If a graph admits such a set coloring where all subsets of X except the empty set are obtained on the edges, then G is called properly set colorable. Ka,b denotes a complete bipartite graph with a and b being the cardinality of the partition sets A,B ⊂ V (G), respectively. An earlier approach to distinguish the edges of a graph by the colors of their adjacent vertices is due to Frank, Harary and Plantholt [2], who introduced the line distinguishing chromatic number of a graph in 1982. Another approach is due to Hopcroft and Krishnamoorthy [4], who established the common notion of harmonious colorings in 1983. Among others Zhang, Liu and Wang [5], and Balister, Riordan and Schelp [1] studied edge colorings of graphs, where the vertices are distinguishable by the set of colors on their incident edges. The concept of set colorings was recently introduced by S.M. Hegde in [3], and describes a way to distinguish edges of a graph by subsets from the color set X. In this paper we will prove his conjecture on proper set colorability of complete bipartite graphs. Therefore, a graph Ka,b without loss of generality with a ≤ b is supposably properly set colorable if and only if it satisfies a = 1 and b = 2n−1 for some positive integer n. Before we prove our result we need some properties and terminology of set colorings of complete bipartite graphs. Therefore, from now on we will consider G to be a complete bipartite graph Ka,b with a proper set coloring f . A necessary condition for f to be a proper set coloring of G with the color set X = {c0, . . . , cn−1} of order n = |X| is a · b = |2X | − 1 = 2 − 1. (1) We define a binary representation binary(i) of a non-negative integer i by the bijective function binary : N0 → {(dm)m∈N0 ∈ {0, 1}N0 | ∃m0 ∈ N ∀m > m0 : dm = 0},