Summary: A note on the degree, size and chromatic index of a uniform
Jeong Han Kim
Let H be a k-uniform hypergraph in which no two edges share more than t common vertices,
and let D denote the maximum degree of a vertex of H. We conjecture that for every > 0,
if D is sufficiently large as a function of t, k and , then the chromatic index of H is at most
(t - 1 + 1/t + )D. We prove this conjecture for the special case of intersecting hypergraphs in
the following stronger form: If H is an intersecting k-uniform hypergraph in which no two edges
share more than t common vertices, and D is the maximum degree of a vertex of H, where D is
sufficiently large as a function of k, then H has at most (t - 1 + 1/t)D edges.
For a k-uniform hypergraph H (which may have multiple edges), let D(H) denote the maximum
degree of a vertex of H, and let (H) denote the chromatic index of H, that is, the minimum
number of colors needed to color the edges of H so that each color class forms a matching. For an
integer t satisfying 1 t k, we say that H is t-simple if every two distinct edges of H have at most
t vertices in common. We propose the following conjecture.
Conjecture 1.1 For every k t 1 and every > 0 there is a finite D0 = D0(k, t, ) so that if
D > D0 then every k-uniform, t-simple hypergraph H with maximum degree at most D satisfies