On the total chromatic number of the direct product of cycles and complete graphs
A k-total coloring of a graph G is an assignment of k colors to the elements (vertices and edges) of G so that adjacent or incident elements have different colors. The total chromatic number is the smallest integer k for which G has a k -total coloring. The well known Total Coloring Conjecture state...
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Published in | R.A.I.R.O. Recherche opérationnelle Vol. 58; no. 2; pp. 1609 - 1632 |
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Main Authors | , , , , , |
Format | Journal Article |
Language | English |
Published |
Paris
EDP Sciences
01.03.2024
|
Subjects | |
Online Access | Get full text |
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Summary: | A
k-total coloring
of a graph
G
is an assignment of
k
colors to the elements (vertices and edges) of
G
so that adjacent or incident elements have different colors. The total chromatic number is the smallest integer
k
for which
G
has a
k
-total coloring. The well known Total Coloring Conjecture states that the total chromatic number of a graph is either Δ(
G
) + 1 (called Type 1) or Δ(
G
) + 2 (called Type 2), where Δ(
G
) is the maximum degree of
G
. We consider the direct product of complete graphs
K
m
×
K
n
. It is known that if at least one of the numbers
m
or
n
is even, then
K
m
×
K
n
is Type 1, except for
K
2
×
K
2
. We prove that the graph
K
m
×
K
n
is Type 1 when both
m
and
n
are odd numbers, by using that the conformable condition is sufficient for the graph
K
m
×
K
n
to be Type 1 when both
m
and
n
are large enough, and by constructing the target total colorings by using Hamiltonian decompositions and a specific color class, called guiding color. We additionally apply our technique to the direct product
C
m
×
K
n
of a cycle with a complete graph. Interestingly, we are able to find a Type 2 infinite family
C
m
×
K
n
, when
m
is not a multiple of 3 and
n
= 2. We provide evidence to conjecture that all other
C
m
×
K
n
are Type 1. |
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ISSN: | 0399-0559 2804-7303 1290-3868 |
DOI: | 10.1051/ro/2024045 |