On the chromatic number of some P5-free graphs

Let G be a graph. We say that 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 $\omega (H[B])<\omega (H)$. We use $P_t$ and $C_t$ to denote a path and a cycle on t vertices, respectively. For two disjoint graphs $F_1$ and $F_2$, we use $F_1\cup F_2$ to denote the graph with vertex set $V(F_1)\cup V(F_2)$ and edge set $E(F_1)\cup E(F_2)$, and use $F_1+F_2$ to denote the graph with vertex set $V(F_1)\cup V(F_2)$ and edge set $E(F_1)\cup E(F_2)\cup \{xy|x\in V(F_1)\text{ and }y\in V(F_2)\}$. In this paper, we prove that (i) $(P_5,C_5,K_{2,3})$-free graphs are perfectly divisible, (ii) $\chi (G)\le 2{\omega }^2(G)?\omega (G)?3$ if G is $(P_5,K_{2,3})$-free with $\omega (G)\ge 2$, (iii) $\chi (G)\le \frac{3}{2}({\omega }^2(G)?\omega (G))$ if G is $(P_5,K_1+2K_2)$-free, and (iv) $\chi (G)\le 3\omega (G)+11$ if G is $(P_5,K_1+(K_1\cup K_3))$-free.     

Point partition numbers: Decomposable and indecomposable critical graphs

Graphs considered in this paper are finite, undirected and loopless, but we allow multiple edges. The point partition number ${\chi }_t(G)$ is the least integer k for which G admits a coloring with k colors such that each color class induces a $(t?1)$-degenerate subgraph of G. So ${\chi }_1$ is the chromatic number and ${\chi }_2$ is the point arboricity. The point partition number ${\chi }_t$ with $t\ge 1$ was introduced by Lick and White. A graph G is called ${\chi }_t$-critical if every proper subgraph H of G satisfies ${\chi }_t(H)<{\chi }_t(G)$. In this paper we prove that if G is a ${\chi }_t$-critical graph whose order satisfies $|G|\le 2{\chi }_t(G)?2$, then G can be obtained from two non-empty disjoint subgraphs $G_1$ and $G_2$ by adding t edges between any pair $u,v$ of vertices with $u\in V(G_1)$ and $v\in V(G_2)$. Based on this result we establish the minimum number of edges possible in a ${\chi }_t$-critical graph G of order n and with ${\chi }_t(G)=k$, provided that $n\le 2k?1$ and t is even. For $t=1$ the corresponding two results were obtained in 1963 by Tibor Gallai.     

On color isomorphic subdivisions

Given a graph H and an integer $k?2$, let $f_k(n,H)$ be the smallest number of colors C such that there exists a proper edge-coloring of the complete graph $K_n$ with C colors containing no k vertex-disjoint color isomorphic copies of H. In this paper, we prove that $f_2(n,H_t)=\mathrm{\Omega }(n^{1+\frac{1}{2t?3}})$ where $H_t$ is the 1-subdivision of the complete graph $K_t$. This answers a question of Conlon and Tyomkyn (2021) [4].     

Large induced subgraphs with three repeated degrees

For any positive integer k, let $C(k)$ denote the least integer such that any n-vertex graph has an induced subgraph with at least $n?C(k)$ vertices, in which at least $\min ?\{k,n?C(k)\}$ vertices are of the same degree. Caro, Shapira and Yuster initially studied this parameter and showed that $\mathrm{\Omega }(k\log ?k)\le C(k)\le {(8k)}^k$. For the first nontrivial case, the authors proved that $3\le C(3)\le 6$, and the exact value was left as an open problem. In this paper, we first show that $3\le C(3)\le 4$, improving the former result as well as a recent result of Kogan. For special families of graphs, we prove that $C(3)=3$ for $K_5$-free graphs, and $C(3)=1$ for large $C_{2s+1}$-free graphs. In addition, extending a result of Erd?s, Fajtlowicz and Staton, we assert that every $K_r$-free graph is an induced subgraph of a $K_r$-free graph in which no degree occurs more than three times.