On Triangle-Free Graphs That Do Not Contain a Subdivision of the Complete Graph on Four Vertices as an Induced Subgraph
نویسندگان
چکیده
We prove a decomposition theorem for the class of triangle-free graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph. We prove that every graph of girth at least 5 in this class is 3-colorable. AMS Classification: 05C75
منابع مشابه
Total Roman domination subdivision number in graphs
A {em Roman dominating function} on a graph $G$ is a function $f:V(G)rightarrow {0,1,2}$ satisfying the condition that every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. A {em total Roman dominating function} is a Roman dominating function with the additional property that the subgraph of $G$ induced by the set of all vertices of positive weight has n...
متن کاملTriangle-free graphs that do not contain an induced subdivision of K4 are 3-colorable
We show that triangle-free graphs that do not contain an induced subgraph isomorphic to a subdivision of K4 are 3-colorable. This proves a conjecture of Trotignon and Vušković [5]. ∗Supported by NSF grant DMS-1550991 and US Army Research Office Grant W911NF16-1-0404. †Partially supported by ANR project Stint under reference ANR-13-BS02-0007 and by the LABEX MILYON (ANR-10-LABX-0070) of Universi...
متن کاملLarge Cliques in C4-Free Graphs
A graph is called C4-free if it contains no cycle of length four as an induced subgraph. We prove that if a C4-free graph has n vertices and at least c1n 2 edges then it has a complete subgraph of c2n vertices, where c2 depends only on c1. We also give estimates on c2 and show that a similar result does not hold for H-free graphs—unless H is an induced subgraph of C4. The best value of c2 is de...
متن کاملChromatic number of ISK4-free graphs
A graph G is said to be ISK4-free if it does not contain any subdivision of K4 as an induced subgraph. In this paper, we propose new upper bounds for chromatic number of ISK4-free graphs and {ISK4, triangle}-free graphs.
متن کاملRooted induced trees in triangle-free graphs
For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of G that is a tree. Further, for a vertex v ∈ V (G), let t(G, v) denote the maximum number of vertices in an induced subgraph of G that is a tree, with the extra condition that the tree must contain v. The minimum of t(G) (t(G, v), respectively) over all connected triangle-free graphs G (and vertices v ∈ V (G)...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Journal of Graph Theory
دوره 84 شماره
صفحات -
تاریخ انتشار 2017