Blue Subgraph Research Articles

Crumby colorings — Red-blue vertex partition of subcubic graphs regarding a conjecture of Thomassen

Thomassen formulated the following conjecture: Every 3-connected cubic graph has a red-blue vertex coloring such that the blue subgraph has maximum degree at most 1 (that is, it consists of a matching and some isolated vertices) and the red subgraph has minimum degree at least 1 and contains no 3-edge path. Since all monochromatic components are small in this coloring and there is a certain irregularity, we call such a coloring crumby. Recently, Bellitto, Klimošová, Merker, Witkowski and Yuditsky [2] constructed an infinite family refuting the above conjecture. Their prototype counterexample is 2-connected, planar, but contains a K4-minor and also a 5-cycle. This leaves the above conjecture open for some important graph classes: outerplanar graphs, K4-minor-free graphs, bipartite graphs. In this regard, we prove that 2-connected outerplanar graphs, subdivisions of K4 and 1-subdivisions of cubic graphs admit crumby colorings. A subdivision of G is genuine if every edge is subdivided at least once. We show that every genuine subdivision of any subcubic multigraph admits a crumby coloring. We slightly generalize some of these results and formulate a few conjectures.

Counterexamples to Thomassen’s Conjecture on Decomposition of Cubic Graphs

We construct an infinite family of counterexamples to Thomassen’s conjecture that the vertices of every 3-connected, cubic graph on at least 8 vertices can be colored blue and red such that the blue subgraph has maximum degree at most 1 and the red subgraph minimum degree at least 1 and contains no path on 4 vertices.

The size Ramsey and connected size Ramsey numbers for matchings versus paths

Given simple graphs F, G, and H, we write F → (G, H) if for every red-blue coloring of the edges of F, there exists either a red subgraph G or a blue subgraph H in F. The size Ramsey number for G and H, denoted by (G, H), is the smallest size of a graph F which satisfies F → (G, H). If in addition F must be connected, then the resulting number is called the connected size Ramsey number for G and H, denoted by (G, H). In this paper, we obtain upper bounds for , and for t ≥ 1 and m ≥ 3. We also determine the exact values of and . In addition, the exact values of for t = 3,4 are given.

On Ramsey (P 3, C 7)-minimal graphs

Let F, G and H be graphs. Notation F → (G, H) means that there is any two-coloring, say red and blue, of all edges of F which contains red subgraph isomorphic to G or blue subgraph isomorphic to H. The graph F is Ramsey (G, H)-minimal if F → (G, H) but F − e ↛ (G, H) for any e ∈ E(F).The class of all Ramsey (G, H)-minimal graphs will be denoted by R(G, H). According to the previous study, we know that graphs in R(P 3, C 7) have at least 13 edges.In this paper, we find graphs in R(P 3, C 7) having seven vertices and 13 and 14 edges, respectively. Further, we give some necessary conditions for Ramsey (P 3, C 7)-minimal graphs with seven vertices.

Unicyclic Ramsey (P 3, Pn )-minimal graphs obtained from trees in the same class

If G, H and F are finite and simple graphs, notation F → (G, H) means that for any red-blue coloring of the edges of F, there is either a red subgraph isomorphic to G or a blue subgraph isomorphic to H. A graph F is a Ramsey (G, H)-minimal graph if F → (G, H) and for every e ∈ E(F), graph F − e ↛ (G, H). The class of all Ramsey (G, H)-minimal graphs (up to isomorphism) will be denoted by R(G, H). The characterization of all graphs in the infinite class R(P 3, Pn ) is still open, for any n ≥ 4. In this paper, we find an infinite families of trees in R(P 3, P 5). We determine how to construct unicyclic graphs in R(P 3, Pn ), for any n ≥ 5 from trees in the same class. Further, we give some properties for the unicyclic graphs constructed from trees in R(P 3, Pn ), for any n ≥ 5.

On Ramsey (P 3, C 6)-minimal graphs for certain order

Let F, G and H be graphs. Notation F → (G, H) means that there is any two-coloring, say red and blue, of all edges of F which contains red subgraph isomorphic to G or blue subgraph isomorphic to H. The graph F is Ramsey (G, H)-minimal graph if F → (G, H) but F − e ↛ (G, H) for any e ∈ E(F). The class of all Ramsey (G, H)-minimal graphs will be denoted by R(G, H). In this paper, we proved that there are only two graphs with six vertices that belong to Ramsey minimal graphs for certain pair of path and cycle (P 3, C 6) and we also determined some graphs with eight vertices in R(P 3, C 6).

The connected size Ramsey number for matchings versus small disconnected graphs

Let F , G , and H be simple graphs. The notation F → ( G , H ) means that if all the edges of F are arbitrarily colored by red or blue, then there always exists either a red subgraph G or a blue subgraph H . The size Ramsey number of graph G and H , denoted by r ( G , H ) is the smallest integer k such that there is a graph F with k edges satisfying F → ( G , H ). In this research, we will study a modified size Ramsey number, namely the connected size Ramsey number. In this case, we only consider connected graphs F satisfying the above properties. This connected size Ramsey number of G and H is denoted by r c ( G , H ). We will derive an upper bound of r c ( n K 2 , H ), n ≥ 2 where H is 2 P m or 2 K 1, t , and find the exact values of r c ( n K 2 , H ), for some fixed n .

Decomposition of cubic graphs related to Wegner’s conjecture

Thomassen formulated the following conjecture: Every 3-connected cubic graph has a red–blue vertex coloring such that the blue subgraph has maximum degree 1 (that is, it consists of a matching and some isolated vertices) and the red subgraph has minimum degree at least 1 and contains no 3-edge path. We prove the conjecture for Generalized Petersen graphs.We indicate that a coloring with the same properties might exist for any subcubic graph. We confirm this statement for all subcubic trees.

Restricted Size Ramsey Number for 2K2 versus Dense Connected Graphs of Order Six

Let G and H be simple graphs. The Ramsey number r(G, H) for a pair of graphs G and H is the smallest number r such that any red-blue coloring of the edges of Kr contains a red subgraph G or a blue subgraph H. The size Ramsey number for a pair of graphs G and H is the smallest number such that there exists a graph F with size satisfying the property that any red-blue coloring of the edges of F contains a red subgraph G or a blue subgraph H. Additionally, if the order of F in the size Ramsey number equals r(G, H), then it is called the restricted size Ramsey number. In 1972, Chvátal and Harary gave the Ramsey number for 2K2 versus any graph H with no isolates. In 1983, Harary and Miller started the investigation of the (restricted) size Ramsey number for some pairs of small graphs with order at most four. In 1983, Faudree and Sheehan continued the investigation and summarized the complete results on the (restricted) size Ramsey number for all pairs of small graphs with order at most four. In 1998, Lortz and Mengenser gave both the size Ramsey number and the restricted size Ramsey number for all pairs of small forests with order at most five. Lately, we investigate the restricted size Ramsey number for 2K2 versus all connected graphs of order five. In this work, we continue the investigation on the restricted size Ramsey number for a pair of small graphs. In particularly, for 2K2 versus dense connected graph of order six.

Connected size Ramsey number for matchings vs. small stars or cycles

The notation $$F\rightarrow (G,H)$$ means that if the edges of F are colored red and blue, then the red subgraph contains a copy of G or the blue subgraph contains a copy of H. The connected size Ramsey number $$\hat{r}_c(G,H)$$ of graphs G and H is the minimum size of a connected graph F satisfying $$F\rightarrow (G,H)$$ . For $$m \ge 2,$$ the graph consisting of m independent edges is called a matching and is denoted by $$mK_2$$ . In 1981, Erdos and Faudree determined the size Ramsey numbers for the pair $$(mK_2, K_{1,t})$$ . They showed that the disconnected graph $$mK_{1,t} \rightarrow (mK_2,K_{1,t})$$ for $$ t,m \ge 1$$ . In this paper, we will determine the connected size Ramsey number $$\hat{r}_c(nK_2, K_{1,3})$$ for $$n\ge 2$$ and $$\hat{r}_c(3K_2, C_4)$$ . We also derive an upper bound of the connected size Ramsey number $$\hat{r}_c(nK_2, C_4),$$ for $$n\ge 4$$ .

Some infinite families of Ramsey ( $${\varvec{P}}_\mathbf{3},{\varvec{P}}_{{\varvec{n}}}$$ P 3 , P n )-minimal trees

For any given two graphs G and H, the notation $$F\rightarrow $$ (G, H) means that for any red–blue coloring of all the edges of F will create either a red subgraph isomorphic to G or a blue subgraph isomorphic to H. A graph F is a Ramsey (G, H)-minimal graph if $$F\rightarrow $$ (G, H) but $$F-e\nrightarrow (G,H)$$ , for every $$e \in E(F)$$ . The class of all Ramsey (G, H)-minimal graphs is denoted by $$\mathcal {R}(G,H)$$ . In this paper, we construct some infinite families of trees belonging to $$\mathcal {R}(P_3,P_n)$$ , for $$n=8$$ and 9. In particular, we give an algorithm to obtain an infinite family of trees belonging to $$\mathcal {R}(P_3,P_n)$$ , for $$n\ge 10$$ .

Restricted size Ramsey number for path of order three versus graph of order five

Let $G$ and $H$ be simple graphs. The Ramsey number for a pair of graph $G$ and $H$ is the smallest number $r$ such that any red-blue coloring of edges of $K_r$ contains a red subgraph $G$ or a blue subgraph $H$. The size Ramsey number for a pair of graph $G$ and $H$ is the smallest number $\hat{r}$ such that there exists a graph $F$ with size $\hat{r}$ satisfying the property that any red-blue coloring of edges of $F$ contains a red subgraph $G$ or a blue subgraph $H$. Additionally, if the order of $F$ in the size Ramsey number is $r(G,H)$, then it is called the restricted size Ramsey number. In 1983, Harary and Miller started to find the (restricted) size Ramsey number for any pair of small graphs with order at most four. Faudree and Sheehan (1983) continued Harary and Miller's works and summarized the complete results on the (restricted) size Ramsey number for any pair of small graphs with order at most four. In 1998, Lortz and Mengenser gave both the size Ramsey number and the restricted size Ramsey number for any pair of small forests with order at most five. To continue their works, we investigate the restricted size Ramsey number for a path of order three versus connected graph of order five.

On partitioning the edges of 1-plane graphs

A 1-plane graph is a graph embedded in the plane such that each edge is crossed at most once. A 1-plane graph is optimal if it has maximum edge density. A red–blue edge coloring of an optimal 1-plane graph G partitions the edge set of G into blue edges and red edges such that no two blue edges cross each other and no two red edges cross each other. We prove the following: (i) Every optimal 1-plane graph has a red–blue edge coloring such that the blue subgraph is maximal planar while the red subgraph has vertex degree at most four; this bound on the vertex degree is worst-case optimal. (ii) A red–blue edge coloring may not always induce a red forest of bounded vertex degree. Applications of these results to graph augmentation and graph drawing are also discussed.

Star-critical Ramsey number of [formula omitted] versus [formula omitted

For two graphs G and H, the Ramsey number r(G,H) is the smallest positive integer r, such that any red/blue coloring of the edges of graph Kr contains either a red subgraph that is isomorphic to G or a blue subgraph that is isomorphic to H. Let Sk=K1,k be a star of order k+1 and Kn⊔Sk be a graph obtained by adding a new vertex v and joining v to k vertices of Kn. The star-critical Ramsey number r∗(G,H) is the smallest positive integer k such that any red/blue coloring of the edges of graph Kr−1⊔Sk contains either a red subgraph that is isomorphic to G or a blue subgraph that is isomorphic to H where r=r(G,H). In this paper, it is shown that r∗(Fn,K4)=4n+2 where n≥4.

On Ramsey [formula omitted]-minimal graphs

Let F be a graph and let G, H denote nonempty families of graphs. We write F→(G,H) if in any 2-colouring of edges of F with red and blue, there is a red subgraph isomorphic to some graph from G or a blue subgraph isomorphic to some graph from H. The graph F without isolated vertices is said to be a (G,H)-minimal graph if F→(G,H) and F−e⁄→(G,H) for every e∈E(F). The set of all (G,H)-minimal graphs (up to isomorphism) is called the Ramsey set ℜ(G,H).We present a procedure, which on the basis of the set of some special graphs, generates an infinite family of (K1,m,G)-minimal graphs, where m≥2 and G is a family of 2-connected graphs. Moreover, graphs obtained by this procedure can be obtained in linear time with respect to their order. In particular, we present how to obtain infinitely many graphs from Ramsey sets ℜ(K1,m,Kn), ℜ(K1,m,Kp,q), for every m,p,q≥2 and n≥3, and from Ramsey sets ℜ(K1,m,Cn), for m≥2 and n∈[4,6]. We show minimal graphs with respect to the number of vertices, which belong to the family ℜ(K1,m,Kn), for m,n≥3. We also present graphs which can be used to the construction of infinitely many graphs belonging to the Ramsey set ℜ(K1,m,G), where m≥2 and G is some family of 2-connected graphs consisting of more than one graph.

Ramsey ($K_{1,2},K_3$)-Minimal Graphs

For graphs $G,F$ and $H$ we write $G\rightarrow (F,H)$ to mean that if the edges of $G$ are coloured with two colours, say red and blue, then the red subgraph contains a copy of $F$ or the blue subgraph contains a copy of $H$. The graph $G$ is $(F,H)$-minimal (Ramsey-minimal) if $G\rightarrow (F,H)$ but $G'\not\rightarrow (F,H)$ for any proper subgraph $G'\subseteq G$. The class of all $(F,H)$-minimal graphs shall be denoted by $R (F,H)$. In this paper we will determine the graphs in $R(K_{1,2},K_3)$.

On Ramsey minimal graphs

For graphs G, F and H we write G→( F, H) to mean that if the edges of G are coloured with two colours, say red and blue, then the red subgraph contains a copy of F or the blue subgraph contains a copy of H. The graph G is ( F, H)- minimal ( Ramsey-minimal) if G→( F, H) but G′↛(F,H) for any proper subgraph G′⊆ G. The class of all ( F, H)-minimal graphs will be denoted by R(F,H) . In this paper we will give two equivalent theorems which characterize the graphs belonging to R(K 1,2,K 1,m) for m⩾3.

The planar Ramsey number for C4 and K5 is 13

The planar Ramsey number PR( G, H) is the smallest integer n such that in any two-colouring (red and blue) of edges of K n such that the red colour induces a planar graph there is a red subgraph G or a blue subgraph H. It is known that Ramsey number for C 4 and K 5 equals to 14. In this paper we show that PR( C 4, K 5)=13.

On fractional Ramsey numbers

The classical Ramsey number r( k, l) is the smallest n such that for any red/blue coloring of the edges of K n either the red subgraph has clique number at least k, or the blue subgraph has clique number at least l. In this paper, we consider a similar parameter — the fractional Ramsey number — in which ‘clique number’ is replaced by ‘fractional clique number’. An exact formula for r f( x, y) is given and some extensions are considered as well.

On the use of senders in generalized ramsey theory for graphs

If F, G and H are graphs, write F → (G, H) to mean that however the edges of F are colored red and blue, either the red (partial) subgraph contains a copy of G or the blue subgraph contains a copy of H. Many interesting questions exist concerning this relation, particularly involving the case in which F is minimal for this property. A useful tool for constructing graphs relevant to such questions, at least when G and H are 3-connected, is developed here, namely graphs called senders. These senders are used to prove a number of theorems about the class of minimal F, as well as various related results. For example, let each of G and H be 3-connected, or a triangle. Then there exists an α > 0 such that if n is sufficiently large, there are at least eαn logn nonisomorphic F such that F → (G, H) in a minimal way.