On-line 3-chromatic graphs - II: Critical graphs

András Gyárfás, Zoltán Király, Jeno Lehel

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13 Citations (Scopus)


On-line coloring of a graph is the following process. The graph is given vertex by vertex (with adjacencies to the previously given vertices) and for the actual vertex a color different from the colors of the neighbors must be irrevocably assigned. The on-line chromatic number of a graph G, χ*(G) is the minimum number of colors needed to color on-line the vertices of G (when it is given in the worst order). A graph G is on-line k-critical if χ*(G1) = k, but χ*(G1) < k for all proper induced subgraphs G1 ⊂ G. We show that there are finitely many (51) connected on-line 4-critical graphs but infinitely many disconnected ones. This implies that the problem whether χ*(G) ≤3 is polynomially solvable for connected graphs but leaves open whether this remains true without assuming connectivity. Using the structure descriptions of connected on-line 3-chromatic graphs we obtain one algorithm which colors all on-line 3-chromatic graphs with 4 colors. It is a tight result. This is a companion paper of [1] in which we analyze the structure of triangle-free on-line 3-chromatic graphs.

Original languageEnglish
Pages (from-to)99-122
Number of pages24
JournalDiscrete Mathematics
Issue number1-3
Publication statusPublished - Dec 1 1997


ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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