Abstract
AbstractLet G be a graph and H a graph possibly with loops. We will say that a graph G is an H-colored graph if and only if there exists a function $$c:E(G)\longrightarrow V(H)$$
c
:
E
(
G
)
⟶
V
(
H
)
. A cycle $$(v_1,\ldots ,v_k,v_1)$$
(
v
1
,
…
,
v
k
,
v
1
)
is an H-cycle if and only if $$(c(v_1 v_2),\ldots ,c(v_{k-1}v_k),$$
(
c
(
v
1
v
2
)
,
…
,
c
(
v
k
-
1
v
k
)
,
$$c(v_kv_1), c(v_1 v_2))$$
c
(
v
k
v
1
)
,
c
(
v
1
v
2
)
)
is a walk in H. Whenever H is a complete graph without loops, an H-cycle is a properly colored cycle. In this paper, we work with an H-colored complete graph, namely G, with local restrictions given by an auxiliary graph, and we show sufficient conditions implying that every vertex in V(G) is contained in an H-cycle of length 3 (respectively 4). As a consequence, we obtain some well-known results in the theory of properly colored walks.
Funder
CONACYT FORDECYT-PRONACES
UNAM-DGAPA-PAPIIT
CONACyT
Publisher
Springer Science and Business Media LLC
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