Triangle-free graphs of tree-width t are ceil((t + 3)/2)-colorable

TL;DR

This paper proves that triangle-free graphs with a given tree-width t can be colored with at most ceil((t + 3)/2) colors, establishing a tight bound and linking it to online coloring of path-width t graphs.

Contribution

The paper introduces a tight upper bound on the chromatic number of triangle-free graphs based on their tree-width, and connects this to online coloring of graphs with bounded path-width.

Findings

Bound of ceil((t + 3)/2) colors for triangle-free graphs of tree-width t.

The bound is proven to be tight.

Connection established between coloring tree-width t graphs and online coloring of path-width t graphs.

Abstract

We prove that every triangle-free graph of tree-width t has chromatic number at most ceil((t + 3)/2), and demonstrate that this bound is tight. The argument also establishes a connection between coloring graphs of tree-width t and on-line coloring of graphs of path-width t.

Equations

$$\max (|V(F^{\prime })|-(c^{\prime }-1)+2,0)+1\le \max (|V(F)|-c^{\prime }+2,1)=|V(F)|-c^{\prime }+2,$$

$$c^{\prime }-\min (|N|,c^{\prime }-1)-\max (t-|N|-c^{\prime }+2,0).$$

Full text

While there exist triangle-free graphs of arbitrarily large chromatic number, forbidding triangles improves bounds on the chromatic number in certain graph classes. For example, planar graphs are $4$-colorable [3, 4] and even deciding their $3$-colorability is an NP-complete problem [6], while all triangle-free planar graphs are $3$-colorable [7]. A graph on $n$ vertices may have chromatic number up to $n$, while Ajtai et al. [1] and Kim [10] proved a tight upper bound O\bigl{(}\sqrt{n/\log n}\bigr{)} on the chromatic number of triangle-free $n$-vertex graphs. The chromatic number of graphs of maximum degree $\Delta$ may be as large as $\Delta+1$, while Johansson [9] proved that triangle-free graphs of maximum degree $\Delta$ have chromatic number $O(\Delta/\log\Delta)$.

On the other hand, no such improvement is possible for graphs with bounded degeneracy. A graph $G$ is $d$-degenerate if each subgraph of $G$ contains a vertex of degree at most $d$. A straightforward greedy algorithm colors every $d$-degenerate graph using $d+1$ colors; and a construction by Blanche Descartes [5] gives for every positive integer $d$ a $d$-degenerate triangle-free graph that is not $d$-colorable, as observed by Kostochka and Nešetřil [11]. A construction of triangle-free graphs with high chromatic number by Zykov [12] also has this property, and the same construction was given (and possibly independently rediscovered) by Alon, Krivelevich, and Sudakov [2].

In this note, we consider the chromatic number of graphs of given tree-width $t$. Note that every graph of tree-width $t$ is $t$-degenerate, but on the other hand, the clique $K_{t+1}$ has tree-width $t$, establishing $t+1$ as the tight upper bound on the chromatic number of graphs of tree-width $t$. We show that the bound can be improved by a constant factor if triangles are forbidden.

We also show that this result is tight. Actually, we give the following stronger result on graphs with bounded clique number. Let $g(t,2)=g(0,k)=1$ for all integers $t\geq 0$ and $k\geq 2$. For $t\geq 1$ and $k\geq 3$, let us inductively define $g(t,k)=\lceil(t+1)/2\rceil+g(\lfloor(t-1)/2\rfloor,k-1)$.

Note that $g(t,3)=\lceil(t+3)/2\rceil$ for all $t\geq 1$ and that g(t,k)>\bigl{(}1-\tfrac{1}{2^{k-2}}\bigr{)}t for all $t\geq 0$ and $k\geq 2$. Theorem 2 motivates the following question, which we were not able to resolve.

Let us remark that in the list coloring setting, the question is much easier to settle—the complete bipartite graph $K_{t,t^{t}}$ has tree-width $t$, but there exists an assignment of lists of size $t$ to its vertices from that it cannot be colored, showing that no improvement analogous to Theorem 1 is possible.

Another natural question concerns graphs with larger girth. The construction of Blanche Descartes [5] actually produces $d$-degenerate graphs of girth six that are not $d$-colorable, and Kostochka and Nešetřil [11] generalize this result to graphs of arbitrarily large girth.

1 Tree-width, path-width, and on-line coloring

To prove Theorems 1 and 2, it is convenient to establish a connection between chromatic number of graphs with given tree-width $t$ and an on-line variant of the chromatic number for graphs of path-width $t$. In on-line coloring [8], the graph to be colored is revealed vertex by vertex and a color has to be assigned to each revealed vertex immediately, with no knowledge regarding the rest of the graph. We need a variation on this idea, revealing the vertices in the order given by a path decomposition of the graph.

Let us first recall some definitions. A tree decomposition of a graph $G$ is a pair $(T,\beta)$, where $T$ is a tree and $\beta$ is a function assigning to each vertex of $T$ a set of vertices of $G$, such that for every $uv\in E(G)$ there exists $z\in V(T)$ with $\{u,v\}\subseteq\beta(z)$, and such that for every $v\in V(G)$, the set $\{z:v\in\beta(z)\}$ induces a non-empty connected subtree in $T$. The width of the decomposition is the maximum of $1+|\beta(z)|$ over all $z\in V(T)$, and the tree-width of $G$ is the minimum possible width of its tree-decomposition. A path decomposition is a tree decomposition $(T,\beta)$ where $T$ is a path, and path-width of $G$ is the minimum possible width of its path decomposition.

Suppose $(P,\beta)$ is a path decomposition of a graph $G$, where $P=z_{0}z_{1}\ldots z_{n}$. We say that the path decomposition is nice if $\beta(z_{0})=\emptyset$ and $|\beta(z_{i})\setminus\beta(z_{i-1})|=1$ for $i=1,\dots,n$. Observe that each path decomposition can be transformed into a nice one of the same width. A nice path decomposition gives a natural way to produce the graph $G$ by adding one vertex at a time: the construction starts with the null graph, and in the $i$-th step for $i=1,\ldots,n$, the unique vertex $v_{i}\in\beta(z_{i})\setminus\beta(z_{i-1})$ is added to $G$ and joined to its neighbors in $\beta(z_{i})$. An on-line coloring algorithm $\mathcal{A}$ at each step of this process assigns a color to the vertex $v_{i}$ distinct from the colors of the neighbors of $v_{i}$ in $\beta(z_{i})$. Note that the assigned color cannot be changed later, and that the algorithm does not in advance know the graph $G$, only its part revealed till the current step of the process. Let $\chi_{\mathcal{A}}(G,P,\beta)$ denote the number of colors $\mathcal{A}$ needs to color $G$ when $G$ is presented to the algorithm $\mathcal{A}$ vertex by vertex according to the nice path decomposition $(P,\beta)$.

Let us remark that we misuse the term “algorithm” a bit, as we are not concerned with the question of efficiency or even computability in any model of computation. Formally, an on-line coloring algorithm $\mathcal{A}$ is a function from the set of triples $(H,Q,\gamma)$, where $H$ is a graph with a nice path decomposition $(Q,\gamma)$, to the integers, such that the following holds. Consider any graph $G$ with a nice path decomposition $(P,\beta)$, where $P=z_{0}z_{1}\ldots z_{n}$ for some $n\geq 1$. For $i=1,\ldots,n$, let $v_{i}$ be the unique vertex in $\beta(z_{i})\setminus\beta(z_{i-1})$, let $P_{i}=P-\{z_{i+1},\ldots,z_{n}\}$, $G_{i}=G-\{v_{i+1},\ldots,v_{n}\}$, and let $\beta_{i}$ be the restriction of $\beta$ to $V(P_{i})$. If $\varphi:V(G)\to\mathbf{Z}$ is defined by $\varphi(v_{i})=\mathcal{A}(G_{i},P_{i},\beta_{i})$, then $\varphi$ is a proper coloring of $G$. And, $\chi_{\mathcal{A}}(G,P,\beta)$ is defined as the number of colors used by this coloring $\varphi$.

Nevertheless, it is easier to think about on-line coloring in the adversarial setting: an enemy is producing the graph $G$ with its nice path decomposition on the fly and the algorithm $\mathcal{A}$ is assigning colors to the vertices as they arrive; and the enemy can construct further parts of the graph depending on the coloring chosen by $\mathcal{A}$ so far. We now present the main result of this section, which we use both to give upper bounds on the chromatic number of graphs with bounded tree-width and to prove the existence of bounded tree-width graphs with large chromatic number.

Lemma 5**.**

Let $t$ and $k$ be positive integers, and let $c$ be the maximum of $\chi(G)$ over all $K_{k}$-free graphs $G$ of tree-width at most $t$. There exists an on-line coloring algorithm $\mathcal{A}_{t}$ such that every $K_{k}$-free graph $H$ that has a nice path decomposition $(P,\beta)$ of width at most $t$ satisfies $\chi_{\mathcal{A}_{t}}(H,P,\beta)\leq c$.

Conversely, if an on-line coloring algorithm $\mathcal{A}^{\prime}_{t}$ satisfies $\chi_{\mathcal{A}^{\prime}_{t}}(H,P,\beta)\leq c^{\prime}$ for every $K_{k}$-free graph $H$ having a path decomposition $(P,\beta)$ of width at most $t$, then $\chi(G)\leq c^{\prime}$ for every $K_{k}$-free graph $G$ of tree-width at most $t$.

Proof.

Let us start with the second claim. Let $G$ be a $K_{k}$-free graph and let $(T,\beta)$ be its tree decomposition of width at most $t$. Without loss of generality, we can assume that the tree $T$ is rooted in a vertex $r$ such that $\beta(r)=\emptyset$ and that each vertex $z\neq r$ of $T$ with parent $z^{\prime}$ satisfies $|\beta(z)\setminus\beta(z^{\prime})|=1$. For each path $P$ in $T$ starting in $r$ and ending in a leaf of $T$, let $G_{P}$ be the subgraph of $G$ induced by $\bigcup_{z\in V(P)}\beta(z)$ and let $\beta_{P}$ be the restriction of $\beta$ to $V(P)$. Then $(P,\beta_{P})$ is a nice path decomposition of $G_{P}$ of width at most $t$, and thus the algorithm $\mathcal{A}^{\prime}_{t}$ can be used to color $G_{P}$ by at most $c^{\prime}$ colors. Furthermore, if $P^{\prime}$ is another root-leaf path in $T$ and some vertex $v$ belongs both to $G_{P}$ and $G_{P^{\prime}}$, then $v\in\beta(z)$ for some vertex $z$ belonging to the shared initial subpath of $P$ and $P^{\prime}$, and thus the algorithm $\mathcal{A}^{\prime}_{t}$ assigns the same color to $v$ in its run on $(G_{P},P,\beta_{P})$ and on $(G_{P^{\prime}},P^{\prime},\beta_{P^{\prime}})$. We conclude that the colorings of the graphs $G_{P}$ for all root-leaf paths $P$ are consistent, and their union gives a proper coloring of $G$ using at most $c^{\prime}$ colors.

Conversely, let $c$ be as given in the statement of the lemma. Consider the “universal” infinite $K_{k}$-free graph $G$ of tree-width at most $t$. That is, $G$ is an infinite graph with rooted tree decomposition $(T,\beta)$ of width at most $t$ satisfying the following conditions. The root $r$ of $T$ has $\beta(r)=\emptyset$. For every vertex $z$ of $T$ and all sets $I\subseteq B\subseteq\beta(z)$ such that $G[I]$ is $K_{k-1}$-free and $|B|\leq t$, there exists a child $z^{\prime}$ of $z$ in $T$ such that $\beta(z^{\prime})=B\cup\{v\}$ for a vertex $v$ not belonging to $\beta(z)$, with $v$ adjacent in $G[\beta(z^{\prime})]$ precisely to the vertices in $I$; and $z$ has no other children. A standard compactness argument shows that there exists a coloring $\varphi$ of $G$ using $c$ colors. Now, for any $K_{k}$-free graph $H$ with a nice path decomposition $(P,\beta)$ of width at most $t$, we can identify $P$ with a path in $T$ and $H$ with an induced subgraph of $G$ in the natural way, and the algorithm $\mathcal{A}_{t}$ works by assigning colors to vertices of $H$ according to the coloring $\varphi$. Hence, $\chi_{\mathcal{A}_{t}}(H,P,\beta)\leq c$. ∎

2 Non-colorability

In this section, we give the construction proving Theorem 2.

Let $H_{0}$ and $H$ be graphs with nice path decompositions $(P_{0},\beta_{0})$ and $(P,\beta)$, respectively. We say that $(H,P,\beta)$ extends $(H_{0},P_{0},\beta_{0})$ if $P_{0}$ is an initial segment of $P$, $\beta_{0}$ is the restriction of $\beta$ to $V(P_{0})$, and $H_{0}$ is an induced subgraph of $H$. Let $z$ be the last vertex of $P$. We say that a coloring $\varphi$ of $H$ is $c$-forced if there exists an independent set $I\subseteq\beta(z)$ of size $c$ such that vertices of $I$ receive pairwise distinct colors according to $\varphi$.

Lemma 6**.**

Let $k\geq 3$ and $c\geq 1$ be integers, and let $\mathcal{A}$ be an on-line coloring algorithm. Let $H_{0}$ be a triangle-free graph with a nice path decomposition $(P_{0},\beta_{0})$ of width at most $2c-2$. Let $c_{0}\leq c-1$ be a non-negative integer. If the coloring $\varphi_{0}$ of $H_{0}$ produced by $\mathcal{A}$ is $c_{0}$-forced, then there exists a triangle-free graph $H$ with a nice path decomposition $(P,\beta)$ of width at most $2c-2$ such that $(H,P,\beta)$ extends $(H_{0},P_{0},\beta_{0})$ and the coloring $\varphi$ of $H$ produced by $\mathcal{A}$ is $(c_{0}+1)$-forced.

Proof.

Let $z_{0}$ be the last vertex of $P_{0}$ and let $I=\{v_{1},\ldots,v_{c_{0}}\}$ be an independent set contained in $\beta_{0}(z_{0})$ such that $\varphi_{0}$ assigns pairwise distinct colors to vertices of $I$. Without loss of generality $\varphi_{0}(v_{1})=1$, …, $\varphi_{0}(v_{c_{0}})=c_{0}$. As $(H,P,\beta)$ will be chosen to extend $(H_{0},P_{0},\beta_{0})$, the coloring $\varphi$ produced by the algorithm $\mathcal{A}$ will match $\varphi_{0}$ on $V(H_{0})$.

We will now append further vertices $x_{1}$, $x_{2}$, …at the end of $P_{0}$ to obtain the path $P$ as follows. Let $\beta(x_{1})=\{v_{1},\ldots,v_{c_{0}},v^{\prime}_{1}\}$ for a new vertex $v^{\prime}_{1}$ with no neighbors, and use the algorithm $\mathcal{A}$ to extend $\varphi$ to this vertex. If $\varphi(v^{\prime}_{1})$ is distinct from $1$, …, $c_{0}$, then we stop the construction. Otherwise, we can by symmetry assume that $\varphi(v^{\prime}_{1})=1$. We then let $\beta(x_{2})=\{v_{1},\ldots,v_{c_{0}},v^{\prime}_{1},v^{\prime}_{2}\}$ for a new vertex $v^{\prime}_{2}$ adjacent to $v_{1}$. If $\varphi(v^{\prime}_{2})$ is distinct from $1$, …, $c_{0}$, then we stop the construction (the independent set receiving $c_{0}+1$ distinct colors is $\{v^{\prime}_{1},v_{2},\ldots,v_{c_{0}},v^{\prime}_{2}\}$). Otherwise, since $\varphi(v^{\prime}_{2})\neq\varphi(v_{1})$, we can assume that $\varphi(v^{\prime}_{2})=2$. Similarly, we proceed for $j=3,\ldots,c_{0}+1$: we set $\beta(x_{j})=\{v_{1},\ldots,v_{c_{0}},v^{\prime}_{1},\ldots,v^{\prime}_{j}\}$, with $v^{\prime}_{j}$ adjacent to $v_{1}$, …, $v_{j-1}$, and depending on the decision of $\mathcal{A}$ regarding the color of $v^{\prime}_{j}$, we either stop with the independent set $\{v^{\prime}_{1},v^{\prime}_{2},\ldots,v^{\prime}_{j-1},v_{j},v_{j-1},\ldots,v_{c_{0}},v^{\prime}_{j}\}$ using $c_{0}+1$ distinct colors, or we can assume that $\varphi(v^{\prime}_{j})=j$. The former happens necessarily at latest when $j=c_{0}+1$.

All the bags created throughout this process have size at most $2c_{0}+1\leq 2c-1$, and thus the width of the resulting path decomposition is at most $2c-2$. ∎

Iterating Lemma 6, we obtain the following.

Corollary 7**.**

Let $t\geq 0$ be an integer. For any on-line coloring algorithm $\mathcal{A}$, there exists a triangle-free graph $H$ with a nice path decomposition $(P,\beta)$ of width at most $t$ such that the coloring $\varphi$ of $H$ produced by $\mathcal{A}$ is $\lceil\tfrac{t+1}{2}\rceil$-forced.

We are now ready to establish Theorem 2.

Proof of Theorem 2.

By Lemma 5, it suffices to show that for every on-line coloring algorithm $\mathcal{A}$, there exists a $K_{k}$-free graph $H$ with a nice path decomposition $(P,\beta)$ of width at most $t$ such that $\chi_{\mathcal{A}}(H,P,\beta)\geq g(t,k)$. We prove the claim by induction on $k$. When $k=2$ or $t=0$, we have $g(t,k)=1$ and the claim obviously holds, with $H$ being a single-vertex graph. Hence, we can assume that $k\geq 3$ and $t\geq 1$. Let $c_{1}=\lceil\tfrac{t+1}{2}\rceil$ and $c_{2}=t-c_{1}=\lfloor\tfrac{t-1}{2}\rfloor$.

Let $H_{0}$ be a triangle-free graph with a nice path decomposition $(P_{0},\beta_{0})$ of width at most $t$ such that the coloring $\varphi_{0}$ of $H_{0}$ produced by $\mathcal{A}$ is $c_{1}$-forced, obtained using Corollary 7. Let $z_{0}$ be the last vertex of $P_{0}$ and let $I$ be an independent set in $H_{0}[\beta_{0}(z_{0})]$ of size $c_{1}$ whose vertices are colored by pairwise distinct colors in $\varphi_{0}$.

Let $\mathcal{A}^{\prime}$ be an on-line coloring algorithm defined as follows: given any graph $H_{1}$ with a nice path decomposition $(P_{1},\beta_{1})$, let $H^{\prime}_{1}$ be the graph obtained from a disjoint union of $H_{0}$ and $H_{1}$ by adding all edges between $I$ and $V(H_{1})$. Let $P^{\prime}_{1}$ be the concatenation of $P_{0}$ and $P_{1}$, with $\beta^{\prime}_{1}(z)=\beta_{0}(z)$ for $z\in V(P_{0})$ and $\beta^{\prime}_{1}(z)=\beta_{1}(z)\cup I$ for $z\in V(P_{1})$. Then $(P^{\prime}_{1},\beta_{1})$ is a nice path decomposition of $H^{\prime}_{1}$. The algorithm $\mathcal{A}^{\prime}$ obtains a coloring of $H_{1}$ as the restriction of the coloring of $H^{\prime}_{1}$ given by the algorithm $\mathcal{A}$ to $V(H_{1})$.

By the induction hypothesis, there exists a $K_{k-1}$-free graph $H_{1}$ with a nice path decomposition $(P_{1},\beta_{1})$ of width at most $t-c_{1}=c_{2}$ such that $\chi_{\mathcal{A}^{\prime}}(H_{1},P_{1},\beta_{1})\geq g(c_{2},k-1)$. Let $H$ and its path decomposition $(P,\beta)$ be chosen as the corresponding graph $H^{\prime}_{1}$ with path decomposition defined as in the previous paragraph. Since $H_{1}$ is $K_{k-1}$-free, $H_{0}$ is triangle-free, and $I$ is an independent set, we conclude that $H$ is $K_{k}$-free. Let $\varphi$ be the coloring of $H$ obtained by $\mathcal{A}$. The restriction of $\varphi$ to $V(H_{0})$ matches $\varphi_{0}$, and in particular $c_{1}$ distinct colors are used on $I$. The restriction of $\varphi$ to $V(H_{1})$ by definition matches the coloring of $H_{1}$ obtained by the algorithm $\mathcal{A}^{\prime}$, and thus it uses at least $g(c_{2},k-1)$ distinct colors. Furthermore, since each vertex of $I$ is adjacent to all vertices of $V(H_{1})$ in $H$, the sets of colors used on $I$ and on $V(H_{1})$ are disjoint. Therefore, $\chi_{\mathcal{A}}(H,P,\beta)\geq c_{1}+g(c_{2},k-1)=g(t,k)$. ∎

3 Colorability

Let $c^{\prime}$ be a positive integer, let $F$ be a graph, and let $\varphi$ be a $c^{\prime}$-coloring of $F$. We say that $\varphi$ is $F$-valid if $F$ does not contain any independent set on which $\varphi$ uses all $c^{\prime}$ distinct colors. We say that a color $a$ is $(F,\varphi)$-forbidden if there exists an independent set $A_{a}\subseteq V(F)$ in $F$ such that $\varphi$ uses all colors except for $a$ on $A_{a}$. We need the following auxiliary claim.

Lemma 8**.**

Let $c^{\prime}$ be a positive integer and let $\varphi$ be a $c^{\prime}$-coloring of a graph $F$. If $\varphi$ is $F$-valid, then at most $\max(|V(F)|-c^{\prime}+2,0)$ colors are $(F,\varphi)$-forbidden.

Proof.

We prove the claim by induction on $|V(F)|$, and thus we assume that Lemma 8 holds for all graphs with fewer than $|V(F)|$ vertices. For each $(F,\varphi)$-forbidden color $a$, let $A_{a}$ be an independent set such that $\varphi$ uses all colors except for $a$ on $A_{a}$.

If $|V(F)|\leq c^{\prime}-2$, then $F$ contains no independent set of size $c^{\prime}-1$, and thus no color is $(F,\varphi)$-forbidden. Hence, suppose that $|V(F)|\geq c^{\prime}-1$, and thus $|V(F)|-c^{\prime}+2\geq 1$. If at most one color is $(F,\varphi)$-forbidden, then the lemma holds. Hence, we can by symmetry assume that colors $1$ and $2$ are forbidden. Note that all $c^{\prime}$ colors appear at least once on $A_{1}\cup A_{2}$. If each $(F,\varphi)$-forbidden color is used on at least two vertices of $F$, then the number of $(F,\varphi)$-forbidden colors is at most $|V(F)|-c^{\prime}$, and the lemma holds.

Hence, we can assume that the color $c^{\prime}$ is $(F,\varphi)$-forbidden and used on exactly one vertex $v$ of $F$. Let $F^{\prime}$ be the graph obtained from $F$ by removing $v$ and all the neighbors of $v$, and let $\varphi^{\prime}$ be the restriction of $\varphi$ to $F^{\prime}$. Note that $\varphi^{\prime}$ is a $(c^{\prime}-1)$-coloring of $F^{\prime}$. We claim that $\varphi^{\prime}$ is $F^{\prime}$-valid. Indeed, if all colors $1$, …, $c^{\prime}-1$ were used on an independent set $A^{\prime}\subseteq V(F^{\prime})$, then all colors $1$, …, $c$ would be used on the independent set $A^{\prime}\cup\{v\}$ in $F$, contradicting the assumption that $\varphi$ is $F$-valid. If a color $a\neq c^{\prime}$ is $(F,\varphi)$-forbidden, then note that $A_{a}$ contains $v$ since $v$ is the only vertex of $F$ of color $c^{\prime}$, and does not contain any of the neighbors of $v$ since $A_{a}$ is an independent set. Hence, $A_{a}\setminus\{v\}$ is an independent set in $F^{\prime}$ on that all colors except for $a$ appear, and thus $a$ is $(F^{\prime},\varphi^{\prime})$-forbidden. Denoting by $f^{\prime}$ the number of $(F^{\prime},\varphi^{\prime})$-forbidden colors, we conclude that at most $f^{\prime}+1$ colors are $(F,\varphi)$-forbidden.

Since $\varphi$ is $F$-valid, the set $A_{c^{\prime}}\cup\{v\}$ is not independent, and thus $v$ has degree at least one. Consequently, $|V(F^{\prime})|\leq|V(F)|-2$. By the induction hypothesis, we conclude that the number of $(F,\varphi)$-forbidden colors is at most

[TABLE]

as required. ∎

We are now ready to bound the chromatic number of triangle-free graphs of tree-width at most $t$.

Proof of Theorem 1.

We need to show that every triangle-free graph $G$ of tree-width at most $t$ can be colored using $c^{\prime}=\lceil(t+3)/2\rceil$ colors. Note that $2c^{\prime}-2>t$. By Lemma 5, it suffices to design an on-line coloring algorithm $\mathcal{A}^{\prime}$ that colors every triangle-free graph $H$ with a nice path decomposition $(P,\beta)$ of width at most $t$ using at most $c^{\prime}$ colors.

The algorithm $\mathcal{A}^{\prime}$ maintains the invariant that the restriction of the $c^{\prime}$-coloring $\varphi$ produced by this algorithm to $\beta(z)$ is $H[\beta(z)]$-valid for every $z\in V(P)$. Consider any vertex $z^{\prime}$ with predecessor $z^{\prime\prime}$ in $P$, let $v$ be the unique vertex in $\beta(z^{\prime})\setminus\beta(z^{\prime\prime})$, and let $N$ be the set of neighbors of $v$ in $\beta(z^{\prime})$. Since $H$ is triangle-free, $N$ is an independent set, and thus at most $\min(|N|,c^{\prime}-1)$ colors appear on $N$ by the invariant. To get a proper coloring maintaining the invariant, it suffices to assign $v$ an arbitrary color that does not appear on $N$ and that is not $(H[\beta(z^{\prime})\setminus(N\cup\{v\})],\varphi)$-forbidden. Since $|\beta(z^{\prime})|\leq t+1$, Lemma 8 implies that the number of such colors is at least

[TABLE]

If $|N|\geq c^{\prime}$, this is at least $c^{\prime}-(c^{\prime}-1)-\max(t-2c^{\prime}+2,0)=1$, using the fact that $2c^{\prime}-2>t$. If $|N|\leq c^{\prime}-1$, this is at least $c^{\prime}-|N|-\max(t-|N|-c^{\prime}+2,0)=\min(2c^{\prime}-t-2,c^{\prime}-|N|)\geq 1$. In either case it is possible to extend the coloring. ∎

Acknowledgments

We would like to thank Endre Csóka, Tomáš Kaiser, and Edita Rollová for fruitful discussions which led to a simplification of the proof of Lemma 8.