Approaches Which Output Infinitely Many Graphs With Small Local Antimagic Chromatic Number · arXivDesk
2009.01996Sep 4, 2020A work that produces infinitely many bipartite graphs with local antimagic chromatic number is 2 or 3, and infinitely many tripartite graphs with local antimagic chromatic number is 3. Many open problems on bipartite and tripartite graphs are suggested
Approaches Which Output Infinitely Many Graphs With Small Local Antimagic Chromatic Number
An edge labeling of a connected graph G=(V,E) is said to be local antimagic if it is a bijection f:E→{1,…,∣E∣}
Nearby in the stack
such that for any pair of adjacent vertices
x
and
y
,
f+(x)=f+(y)
, where the induced vertex label
f+(x)=∑f(e)
, with
e
ranging over all the edges incident to
x
. The local antimagic chromatic number of
G
, denoted by
χla(G)
, is the minimum number of distinct induced vertex labels over all local antimagic labelings of
G
. In this paper, we (i) give a sufficient condition for a graph with one pendant to have
χla≥3
. A necessary and sufficient condition for a graph to have
χla=2
is then obtained; (ii) give a sufficient condition for every circulant graph of even order to have
χla=3
; (iii) construct infinitely many bipartite and tripartite graphs with
χla=3
by transformation of cycles; (iv) apply transformation of cycles to obtain infinitely many one-point union of regular (possibly circulant) or bi-regular graphs with
χla=2,3
. The work of this paper suggests many open problems on the local antimagic chromatic number of bipartite and tripartite graphs.