Excluding the fork and antifork

Chudnovsky, Maria; Cook, L. F.; Seymour, Paul

doi:10.1016/j.disc.2019.111786

Cited by 6 publications

(7 citation statements)

References 2 publications

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“…Therefore, x i has no nonneighbor in N (A), which means that X i is complete to N (A). This proves (2).…”

Section: Lemma 22 [8]supporting

confidence: 53%

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Perfect divisibility and coloring of some fork-free graphs

Wu¹,

Xu²

2023

Preprint

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A hole is an induced cycle of length at least four, and an odd hole is a hole of odd length. A fork is a graph obtained from K 1,3 by subdividing an edge once. An odd balloon is a graph obtained from an odd hole by identifying respectively two consecutive vertices with two leaves of K 1,3 . A gem is a graph that consists of a P 4 plus a vertex adjacent to all vertices of the P 4 . A butterfly is a graph obtained from two traingles by sharing exactly one vertex. A graph G is perfectly divisible if for each induced subgraph H of G, V (H) can be partitioned into A and B such that H[A] is perfect and ω(H[B]) < ω(H). In this paper, we show that (odd balloon, fork)-free graphs are perfectly divisible (this generalizes some results of Karthick et al). As an application, we show that χ(G) ≤ ω(G)+1 2 if G is (fork, gem)-free or (fork, butterfly)-free.

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“…Therefore, x i has no nonneighbor in N (A), which means that X i is complete to N (A). This proves (2).…”

Section: Lemma 22 [8]supporting

confidence: 53%

“…In [2,3,11], the authors proved that (fork, H)-free graphs are linearly χ-bounded when H ∈ {C 4 , K 4 , diamond, K 3 ∪ K 1 , paw, P 3 ∪ 2K 1 , K 5 − e, antifork}, where an antifork is the complement graph of a fork.…”

Section: Introductionmentioning

confidence: 99%

Perfect divisibility and coloring of some fork-free graphs

Wu¹,

Xu²

2023

Preprint

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show abstract

“…The second author with Maffray [8] showed that every (

{P}_{4}+{P}_{1},\bar{{P}_{4}+{P}_{1}}

)‐free graph

G

satisfies

\chi (G)\le \lceil \phantom{\rule[-0.5em]{}{0ex}}\frac{5\omega (G)}{4}\rceil

, and that the bound is tight. Recently, Chudnovsky et al [2] showed that every (fork, anti‐fork)‐free graph

G

satisfies

\chi (G)\le 2\omega (G)

, and that the bound is asymptotically tight. Thus, Problem 1 is open for seven pairwise nonisomorphic forests on five vertices, as there are 10 such forests.…”

Section: Introductionmentioning

confidence: 99%

Optimal chromatic bound for (P2+P3,P2+P3¯ ${P}_{2}+{P}_{3},\bar{{P}_{2}+{P}_{3}}$)‐free graphs

Char

Thiyagarajan

2023

Journal of Graph Theory

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“…• Recently, Chudnovsky et al [4] proved a structure theorem for the class of (fork, antifork)-free graphs, and used it to prove that every (fork, antifork)-free graph G satisfies χ(G) 2ω(G). (Here, an antifork is the complement graph of a fork.…”

Section: Introductionmentioning

confidence: 99%

Coloring Graph Classes with no Induced Fork via Perfect Divisibility

Karthick¹,

Kaufmann²,

Sivaraman

2022

Electron. J. Combin.

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For a graph $G$, $\chi(G)$ will denote its chromatic number, and $\omega(G)$ its clique number. A graph $G$ is said to be perfectly divisible if for all induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A$, $B$ such that $H[A]$ is perfect and $\omega(H[B]) < \omega(H)$. An integer-valued function $f$ is called a $\chi$-binding function for a hereditary class of graphs $\cal C$ if $\chi(G) \leq f(\omega(G))$ for every graph $G\in \cal C$. The fork is the graph obtained from the complete bipartite graph $K_{1,3}$ by subdividing an edge once. The problem of finding a quadratic $\chi$-binding function for the class of fork-free graphs is open. In this paper, we study the structure of some classes of fork-free graphs; in particular, we study the class of (fork, $F$)-free graphs $\cal G$ in the context of perfect divisibility, where $F$ is a graph on five vertices with a stable set of size three, and show that every $G\in \cal G$ satisfies $\chi(G)\le \omega(G)^2$. We also note that the class $\cal G$ does not admit a linear $\chi$-binding function.

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Excluding the fork and antifork

Cited by 6 publications

References 2 publications

Perfect divisibility and coloring of some fork-free graphs

Perfect divisibility and coloring of some fork-free graphs

Optimal chromatic bound for (P2+P3,P2+P3¯ ${P}_{2}+{P}_{3},\bar{{P}_{2}+{P}_{3}}$)‐free graphs

Coloring Graph Classes with no Induced Fork via Perfect Divisibility

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