**Definition.** Suppose $H=(V,E)$ is a hypergraph. Call a hyperedge $e=\{v_1,v_2,\dotsc,v_k\}$ a **$k$-star** if in the 1-skeleton of $H$ there is a copy of $K^{1,k-1}$ whose union equals $e$. (Obviously in the case $k=2$, $e$ is an usual edge.)

**EDITED:** Consider the following two properties:

(a) For every hyperedge $e$ of $H$ there exists $k\in\mathbb{N}$ such that $e$ is $k$-star.

(b) If two hyperedges $e_i$ and $e_j$ which share at least one vertices ($|e_i\cap e_j|\geq1$) then there is an usual edge that connect internal nodes of this two hyperedges.

Question:Does anybody know how to prove or disprove the following guess about edge coloring of Hypergraphs?

Let $\Delta(H)$ denote the maximum of vertex-degrees of a finite hypergraph $H$.

Guess.Let $H$ be a simple hypergraph satisfying (a) and (c). Then there exists an $(\Delta+1)$-edge-coloring so that any two edges which share one vertices have distinct colors.

need not be unique, so (c) is simplyundefined. Would you please think about what you are asking and then formulate the question unambiguously? $\endgroup$ – Peter Heinig Apr 5 '18 at 11:32moredifficult, and in particular, foranyhypergraph $H$ once can easily 'saturate' $H$ with enough 2-element hyperedges to make (a) and (b) while not lowering the chromatic number. The condition you now mentioned makes more sense, since it seems to edge-coloring easier. By the way, do you allowmultiple edges'in your 'hypergraphs"? (some peapole do, some don't.) $\endgroup$ – Peter Heinig Apr 6 '18 at 7:33