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URN etd-1223109-135602
Author Nicolas Roussel
Author's Email Address No Public.
Statistics This thesis had been viewed 5061 times. Download 1718 times.
Department Applied Mathematics
Year 2009
Semester 1
Degree Ph.D.
Type of Document
Language English
Title Circular colorings and acyclic choosability of graphs
Date of Defense 2009-12-14
Page Count 122
Keyword
  • Circular coloring
  • total coloring
  • missing cycles
  • maximum average degree
  • planar graphs
  • acyclic choosability
  • circular (r;1)-total labeling
  • (d;1)-total labeling
  • Abstract Abstract: This thesis studies five kinds of graph colorings: the circular coloring,
    the total coloring, the (d; 1)-total labeling, the circular (r; 1)-total labeling, and the
    acyclic list coloring.
    We give upper bounds on the circular chromatic number of graphs with small
    maximum average degree, mad for short. It is proved that if mad(G)<22=9 then G
    has a 11=4-circular coloring, if mad(G) < 5=2 then G has a 14=5-circular coloring.
    A conjecture by Behzad and Vizing implies that Δ+2 colors are always sufficient
    for a total coloring of graphs with maximum degree Δ. The only open case for planar graphs is for Δ = 6. Let G be a planar in which no vertex is contained in cycles of all lengths between 3 and 8. If Δ(G) = 6, then G is total 8-colorable. If Δ(G) = 8, then G is total 9-colorable.
    Havet and Yu [23] conjectured that every subcubic graph G ̸=K4 has (2; 1)-total
    number at most 5. We confirm the conjecture for graphs with maximum average
    degree less than 7=3 and for flower snarks.
    We introduce the circular (r; 1)-total labeling. As a relaxation of the aforementioned
    conjecture, we conjecture that every subcubic graph has circular (2; 1)-total number at most 7. We confirm the conjecture for graphs with maximum average degree less than 5=2.
    We prove that every planar graph with no cycles of lengths 4, 7 and 8 is acyclically
    4-choosable. Combined with recent results, this implies that every planar
    graph with no cycles of length 4;k; l with 5 6 k < l 6 8 is acyclically 4-choosable.
    Advisory Committee
  • Hong-Gwa Yeh - chair
  • Mickael Montassier - co-chair
  • Gerard Chang - co-chair
  • Ko-Wei Lih - co-chair
  • Tsai-Lien Wong - co-chair
  • Yeong-Nan Yeh - co-chair
  • Li-Da Tong - co-chair
  • André Raspaud - advisor
  • Xuding Zhu - advisor
  • Files
  • etd-1223109-135602.pdf
  • indicate access worldwide
    Date of Submission 2009-12-23

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