PC-Trees and Planar Graphs

8
On the strong edge coloring of Halin graphs
AUTHOR: Hsin-Hao Lai 1 , Ko-Wei Lih 2 , and Ping-Ying Tsai 3 蔡秉穎
1 Department of Mathematics, National Kaohsiung Normal University, Yanchao, Kaohsiung
824, Taiwan
hsinhaolai@nknucc.nknu.edu.tw
2 Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan
makwlih@sinica.edu.tw
3 Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan
bytsai@math.sinica.edu.tw
ABSTRACT
A proper edge coloring of a graph G is an assignment of colors to the edges of G such
that no two edges with a common vertex receive the same color. A strong edge coloring
of G is a proper edge coloring, with the further condition that no two edges with the
same color lay on a path of length three. The strong chromatic index of G, denoted by
sχ ′ (G), is the minimum number of colors needed for a strong edge coloring of G. A Halin
graph G = T ∪ C is a plane graph constructed from a tree T without vertices of degree
2 by connecting all leaves through a cycle C such that C is the boundary of the exterior
face. Previously, Shiu et al. have studied the strong chromatic index of some classes of
cubic Halin graphs. They also conjectured that sχ ′ (G) ≤ sχ ′ (T ) + 4 for all Halin graphs.
In this talk, we settle this conjecture by proving a stronger result stated as follows:
If a Halin graph G = T ∪ C is different from a certain necklace Ne2 and any wheel Wn
with n ̸≡ 0 (mod 3), then we have sχ ′ (G) ≤ sχ ′ (T ) + 3.
KEYWORDS: Strong edge coloring, strong chromatic index, Halin graph.