Questions on Color-Critical Subgraphs

Abstract

In our work, we define a k-tuple of positive integers $(x_1, \ldots , x_k)$ to be a $\chi$-sequence if there exists a k-chromatic graph G such that for each $i \in \{1, \ldots , k\}$, the order of a minimum i-chromatic subgraph of G is equal to $x_i$. Denote by $\mathcal {X}_k$ the set of all $\chi$-sequences of length k. A very difficult question is to determine, for a given $(x_1, \ldots , x_k) \in \mathcal {X}_k$, the set of all integers y such that $(x_1, \ldots , x_k, y) \in \mathcal {X}_{k+1}$. We propose a few variants of this question and elaborate upon a number of partial results along the way.