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Some results on total weight choosability

T. Wu, J. Luo, Y. Gao

Abstract

A graph $G=(V,E)$ is called $(k,k')$-choosable if for any total list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and assigns to each edge $e$ a set $L(e)$ of $k'$ real numbers, there is a mapping $f:V\cup E\rightarrow \mathbb{R}$ such that $f(y)\in L(y)$ for any $y\in V\cup E$ and for any two adjacent vertices $v, v'$, $\sum_{e\in E(v)}f(e)+f(v)\neq \sum_{e\in E(v')}f(e)+f(v')$, where $E(x)$ denotes the set of incident edges of a vertex $x\in V(G)$. In this paper, we characterize a sufficient condition on $(1,2)$-choosable of graphs. We show that every connected $(n,m)$-graph is both $(2,2)$-choosable and $(1,3)$-choosable if $m=n$ or $n+1$, where $(n,m)$-graph denotes the graph with $n$ vertices and $m$ edges. Furthermore, we prove that some graphs obtained by some graph operations are $(2,2)$-choosable.

Some results on total weight choosability

Abstract

A graph is called -choosable if for any total list assignment which assigns to each vertex a set of real numbers, and assigns to each edge a set of real numbers, there is a mapping such that for any and for any two adjacent vertices , , where denotes the set of incident edges of a vertex . In this paper, we characterize a sufficient condition on -choosable of graphs. We show that every connected -graph is both -choosable and -choosable if or , where -graph denotes the graph with vertices and edges. Furthermore, we prove that some graphs obtained by some graph operations are -choosable.
Paper Structure (4 sections, 29 theorems, 22 equations, 1 figure)

This paper contains 4 sections, 29 theorems, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A solution to Problem \ref{['art15']}
  3. Total weight choosability of $(n,m)$-graphs when $m=n$ and $n+1$
  4. Total weight choosability of some graphs under some graph decorations

Key Result

Lemma 2.1

(chang) If ${\rm per}(B_{G})\neq 0$. Then $G$ is $(1,2)$-choosable.

Figures (1)

  • Figure 1: Bicyclic graphs $B_1(p,q)$, $B_2(p,q,r)$ and $B_3(p,q,r)$

Theorems & Definitions (47)

  • Conjecture 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • ...and 37 more