Supermodular Extension of Vizing's Edge-Coloring Theorem

Abstract

Kőnig's edge-coloring theorem for bipartite graphs and Vizing's edge-coloring theorem for general graphs are celebrated results in graph theory and combinatorial optimization. Schrijver generalized Kőnig's theorem to a framework defined with a pair of intersecting supermodular functions. The result is called the supermodular coloring theorem. This paper presents a common generalization of Vizing's theorem and a weaker version of the supermodular coloring theorem. To describe this theorem, we introduce intersecting 2/3-supermodular functions, which are extensions of intersecting supermodular functions. The paper also provides an alternative proof of Gupta's edge-coloring theorem using a special case of this supermodular version of Vizing's theorem.

Figures (4)

• Figure 1: The relationship between the coloring-type theorems. The arrows mean implications.
• Figure 2: A maximal trail $P$ satisfying the condition (\ref{['P_cd1']}), and that satisfying the condition (\ref{['P_cd2']}).
• Figure 3: A path $P$ in Step 1. of the edge-orientation algorithm.
• Figure 4: A maximal sequence $\{(Y_0,u_0),\ldots,(Y_l,u_l)\}$ and a maximal sequence $\{x_0,\ldots,x_p\}$.

Theorems & Definitions (40)

• Theorem 1.1: Kőnig konig1916
• Theorem 1.2: Vizing vizing1965
• Theorem 1.3: Gupta gupta1978
• Theorem 1.4: Gupta gupta1974
• Theorem 1.5: Gupta gupta1974
• Theorem 1.6
• Theorem 1.7: Schrijver schrijver1985
• Theorem 1.8: Schrijver schrijver1985
• Theorem 1.9
• Theorem 1.10