The Relative Oriented Clique Number of Triangle-Free Planar Graphs Is 10

Das, Soura Sena; Nandi, Soumen; Sen, Sagnik

doi:10.1007/978-3-030-39219-2_22

Cited by 1 publication

(4 citation statements)

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“…As mentioned before, Das, Nandi and Sen [13] solved Conjecture 1.5 for (n, m) = (1, 0). Moreover, they noted that the solution for (n, m) = (0, 2) is similar.…”

Section: Our Contributionsmentioning

confidence: 77%

“…On the other hand, it turns out, even though finding clique numbers for these two families, that is, the families of planar and triangle-free planar graphs, is trivial for (n, m) = (0, 1), and it is quite a challenging problem for the other values of (n, m). There have been dedicated studies to explore these parameters even for (n, m) = (1, 0) [12,13], and later for general values [4,9].…”

Section: Context and Motivationmentioning

confidence: 99%

“…In the meantime, Sopena [37] enquired the exact values of ω r(1,0) (P 3 ) and ω r(1,0) (P 4 ) in his survey in 2016 (latest to date) on oriented coloring as open problems. Das, Nandi and Sen [13] solved the second problem by showing ω r(1,0) (P 4 ) = 10 while the first question remains open still.…”

Section: Context and Motivationmentioning

confidence: 99%

“…• For oriented graphs, that is, for (n, m) = (1, 0), a number of papers dealt with the notions of absolute and relative cliques: [5,12,13,15,21,30]. A similar variant of homomorphisms of oriented graphs, called pushable homomorphisms of oriented graphs, also studies the notions of absolute and relative cliques [7].…”

Section: The Significance Of Relative Cliquesmentioning

confidence: 99%

See 3 more Smart Citations

On Relative Clique Number of Triangle-Free Planar Colored Mixed Graphs

Nandi¹,

Sen

Taruni

2022

Lecture Notes in Computer Science

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An (n, m)-graph is a graph with n types of arcs and m types of edges. A homomorphism of an (n, m)-graph G to another (n, m)-graph H is a vertex mapping that preserves the adjacencies along with their types and directions. The order of a smallest (with respect to the number of vertices) such H is the (n, m)-chromatic number of G. Moreover, an (n, m)-relative clique R of an (n, m)-graph G is a vertex subset of G for which no two distinct vertices of R get identified under any homomorphism of G. The (n, m)-relative clique number of G, denoted by ω r(n,m) (G), is the maximum |R| such that R is an (n, m)-relative clique of G. In practice, (n, m)relative cliques are often used for establishing lower bounds of (n, m)-chromatic number of graph families.Generalizing an open problem posed by Sopena [Discrete Mathematics 2016] in his latest survey on oriented coloring, Chakroborty, Das, Nandi, Roy and Sen [Discrete Applied Mathematics 2022] conjectured that ω r(n,m) (G) ≤ 2(2n + m) 2 + 2 for any triangle-free planar (n, m)-graph G and that this bound is tight for all (n, m) = (0, 1). In this article, we positively settle this conjecture by improving the previous upper bound of ω r(n,m) (G) ≤ 14(2n+m) 2 +2 to ω r(n,m) (G) ≤ 2(2n+m) 2 +2, and by finding examples of triangle-free planar graphs that achieve this bound. As a consequence of the tightness proof, we also establish a new lower bound of 2(2n + m) 2 + 2 for the (n, m)-chromatic number for the family of triangle-free planar graphs.

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“…As mentioned before, Das, Nandi and Sen [13] solved Conjecture 1.5 for (n, m) = (1, 0). Moreover, they noted that the solution for (n, m) = (0, 2) is similar.…”

Section: Our Contributionsmentioning

confidence: 77%