Bor-Liang Chen

Equitable coloring and the maximum degree

Abstract Let Δ(G) denote the maximum degree of a graph G . The equitable Δ-coloring conjecture asserts that a connected graph G is equitably Δ( G )-colorable if it is different from K m , C 2 m + 1 and K 2 m + 1,2 m + 1 for all m ⩾ 1. This conjecture is established for graphs G satisfying Δ( G ) ⩾ | G |/2 or Δ( G ) ⩽ 3.

Equitable coloring of trees

Abstract A graph is equitably k -colorable if its vertices can be partitioned into k independent sets of as near equal sizes as possible. Regarding a non-null tree T as a bipartite graph T ( X , Y ), we show that T is equitably k -colorable if and only if (i) k ≥ 2 when | | X | − | Y | | ≤ 1; (ii) k ≥ max{3, ⌈(| T | + 1)/(α( T − N [ v ]) + 2)⌉} when | | X | − | Y | | > 1. In case (ii), v is an arbitrary vertex of maximum degree in T and the number α( T − N [ v ]) denotes the independence number of the subgraph of T obtained by deleting v and all its adjacent vertices.

Hamiltonian uniform subset graphs

The uniform subset graph G(n, k, t) is defined to have all k-subsets of an n-set as vertices and edges joining k-subsets intersecting at t elements. We conjecture that G(n, k, t) is hamiltonian when it is different from the Petersen graph and does possess cycles. We verify this conjecture for k − t = 1, 2, 3 and for suitably large n when t = 0, 1.

Equitable and m-Bounded Coloring of Split Graphs

An equitable coloring of a graph is a proper coloring such that the sizes of color classes are as even as possible. An m-bounded coloring of a graph is a proper coloring such that the sizes of color classes are all bounded by a preassigned number m. Formulas for the equitable and m-bounded chromatic numbers of a split graph are established in this paper. It is proved that split graphs satisfy the equitable Δ-coloring conjecture in Chen, Lih and Wu [4].

Diameters of iterated clique graphs of chordal graphs

The clique graph K(G) of a graph is the intersection graph of maximal cliques of G. The iterated clique graph Kn(G) is inductively defined as K(Kn−1(G)) and K1(G) = K(G). Let the diameter diam(G) be the greatest distance between all pairs of vertices of G. We show that diam(Kn(G)) = diam(G) — n if G is a connected chordal graph and n ≤ diam(G). This generalizes a similar result for time graphs by Bruce Hedman.

A note on the m -bounded chromatic number of a tree

Abstract The m-bounded chromatic number of a graph G is the smallest number of colors required for a proper coloring of G in which each color is used at most m times. We will establish an exact formula for the m -bounded chromatic number of a tree.

Equivalence of two conjectures on equitable coloring of graphs

A graph G is said to be equitably k-colorable if there exists a proper k-coloring of G such that the sizes of any two color classes differ by at most one. Let Δ(G) denote the maximum degree of a vertex in G. Two Brooks-type conjectures on equitable Δ(G)-colorability have been proposed in Chen and Yen (Discrete Math., 2011) and Kierstead and Kostochka (Combinatorica 30:201–216, 2010) independently. We prove the equivalence of these conjectures.