The degeneracy and Alon-Tarsi number under $F$-sum operations

Abstract

The Alon-Tarsi number of a graph $ G $ is the smallest $ k $ such that there exists an orientation $ D $ of $ G $ with maximum outdegree $ k - 1 $ satisfying that the number of even Eulerian subgraphs is different from the number of odd Eulerian subgraphs. The degeneracy of a graph $ G $ is the maximum value of the minimum degree over all subgraphs of $ G $. In this paper, we obtain a characterization of graphs with $AT(G)=2$ for any graph $G$, and study the Alon-Tarsi number of $F$-sum in terms of degeneracy.

Figures (14)

• Figure 1: $P_4$ and $S(P_4)$, $R(P_4)$, $Q(P_4)$, $T(P_4).$
• Figure 2: $P_2,C_3$ and $P_2 \square C_3$ .
• Figure 3: $P_4 {+}_{F} P_3$.
• Figure 4: $\Theta_{2,2,2}$ and $\Theta_{2,2,4}$.
• Figure 5: $C_3 {+}_{S} C_3$.

Theorems & Definitions (17)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3