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Register a new userSome Results on $k$-Critical $P_5$-Free Graphs
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Added by
Shenwei Huang
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic number less than $k$. The study of $k$-vertex-critical graphs for graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there is a polynomial-time algorithm to decide if a graph in the class is $(k-1)$-colorable. In this paper, we prove that for every fixed integer $kge 1$, there are only finitely many $k$-vertex-critical ($P_5$,gem)-free graphs and $(P_5,overline{P_3+P_2})$-free graphs. To prove the results we use a known structure theorem for ($P_5$,gem)-free graphs combined with properties of $k$-vertex-critical graphs. Moreover, we characterize all $k$-vertex-critical ($P_5$,gem)-free graphs and $(P_5,overline{P_3+P_2})$-free graphs for $k in {4,5}$ using a computer generation algorithm.
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Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the path on $t$ vertices. A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic number less than $k$. The study of $k$-vertex-critical graphs for graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there is a polynomial-time algorithm to decide if a graph in the class is $(k-1)$-colorable. In this paper, we initiate a systematic study of the finiteness of $k$-vertex-critical graphs in subclasses of $P_5$-free graphs. Our main result is a complete classification of the finiteness of $k$-vertex-critical graphs in the class of $(P_5,H)$-free graphs for all graphs $H$ on 4 vertices. To obtain the complete dichotomy, we prove the finiteness for four new graphs $H$ using various techniques -- such as Ramsey-type arguments and the dual of Dilworths Theorem -- that may be of independent interest.
For a graph $H$ and a $k$-chromatic graph $F,$ if the Turan graph $T_{k-1}(n)$ has the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs (for $n$ large enough), then $H$ is called $F$-Turan-good, or $k$-Turan-good for short if $F$ is $K_k.$ In this paper, we construct some new classes of $k$-Turan-good graphs and prove that $P_4$ and $P_5$ are $k$-Turan-good for $kge4.$
The $k$-power domination problem is a problem in graph theory, which has applications in many areas. However, it is hard to calculate the exact $k$-power domination number since determining k-power domination number of a generic graph is a NP-complete problem. We determine the exact $k$-power domination number in two graphs which have the same number of vertices and edges: pseudofractal scale-free web and Sierpinski gasket. The $k$-power domination number becomes 1 for $kge2$ in the Sierpinski gasket, while the $k$-power domination number increases at an exponential rate with regard to the number of vertices in the pseudofractal scale-free web. The scale-free property may account for the difference in the behavior of two graphs.
The Borodin-Kostochka Conjecture states that for a graph $G$, if $Delta(G) geq 9$ and $omega(G) leq Delta(G)-1$, then $chi(G)leqDelta(G) -1$. We prove the Borodin-Kostochka Conjecture for $(P_5, text{gem})$-free graphs, i.e., graphs with no induced $P_5$ and no induced $K_1vee P_4$.
A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph has a bisimplicial vertex. Both Hadwigers conjecture and the ErdH{o}s-Lovasz Tihany conjecture have been shown to be true for quasi-line graphs, but are open for even-hole-free graphs. In this note, we prove that for all $kge7$, every even-hole-free graph with no $K_k$ minor is $(2k-5)$-colorable; every even-hole-free graph $G$ with $omega(G)<chi(G)=s+t-1$ satisfies the ErdH{o}s-Lovasz Tihany conjecture provided that $ tge s> chi(G)/3$. Furthermore, we prove that every $9$-chromatic graph $G$ with $omega(G)le 8$ has a $K_4cup K_6$ minor. Our proofs rely heavily on the structural result of Chudnovsky and Seymour on even-hole-free graphs.
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