14-Vertex Graph Coloring Game

• 14-vertex graph coloring games are finite challenges that present multiple distinct models, such as Painter–Builder, online token, slow-coloring, and eternal recoloring, each enforcing proper coloring under adversarial conditions.
• They analyze variant-specific invariants—like minimum palette size, cumulative score, and convergence time—demonstrating that structural features, not just vertex count, drive key results.
• Researchers deploy combinatorial and probabilistic strategies to establish rigorous bounds and gameplay dynamics, highlighting the sensitivity of outcomes to underlying graph configurations.

Searching arXiv for the cited graph-coloring game papers to ground the article. A 14-vertex graph coloring game is a finite graph-coloring game played on a graph of order $14$, but the literature uses this label for several distinct models: adversarial one-shot coloring, online token coloring, score-based online painting, eternal recoloring, and synchronous distributed coloring. Across these models, the key parameters are not determined by order alone. In the Painter–Builder game on $14$ vertices, the rigorous bounds are $2 \le k_{\min}(14)\le 5$; in the online token graph model, bipartite $14$-vertex host graphs satisfy $f(w,G)=w$; in the slow-coloring game, $s(P_{14})=21$ and $s(K_{14})=105$; and in the eternal A-game, $\chi_g^{\infty}(P_{14})=\chi_g^{\infty}(C_{14})=4$ (Bednarska-Bzdęga et al., 2016, Milans et al., 2017, Mahoney et al., 2015, Klostermeyer et al., 2018).

1. Model families and invariant structure

The cited literature studies several non-equivalent coloring games on $14$-vertex graphs. In the Painter–Builder game, Painter colors vertices while Builder adds edges, with properness maintained at all times. In the online token graph model, Spoiler places tokens on vertices of a fixed host graph $G$ and Algorithm must immediately color them while maintaining token-graph properness under a width bound $14$0. In the slow-coloring game, Lister marks vertices and Painter colors an independent subset, with the objective measured by the sum-color cost $14$1. In the eternal game chromatic framework, every vertex is colored or recolored once per round and the goal is to maintain properness forever. In the network coloring game, all vertices act synchronously and choose colors from local information only. In the standard graph coloring game, Alice and Bob alternately color previously uncolored vertices and the main invariant is the game chromatic number $14$2; related work studies indicated and independence variants, as well as game-vertex-criticality (Bednarska-Bzdęga et al., 2016, Milans et al., 2017, Mahoney et al., 2015, Klostermeyer et al., 2018, Fryganiotis et al., 2021, Bradshaw, 2020, Jakovac et al., 2021).

Model	Principal invariant	Representative 14-vertex specialization
Painter–Builder proper coloring	$14$3	$14$4
Online token graph coloring	$14$5	$14$6 for bipartite, hence online-perfect, $14$7-vertex graphs
Slow-coloring game	$14$8	$14$9, $2 \le k_{\min}(14)\le 5$0, $2 \le k_{\min}(14)\le 5$1
Eternal game coloring	$2 \le k_{\min}(14)\le 5$2	$2 \le k_{\min}(14)\le 5$3, $2 \le k_{\min}(14)\le 5$4, $2 \le k_{\min}(14)\le 5$5
Network coloring game	convergence time $2 \le k_{\min}(14)\le 5$6	with $2 \le k_{\min}(14)\le 5$7, $2 \le k_{\min}(14)\le 5$8 with high probability
Standard game and criticality	$2 \le k_{\min}(14)\le 5$9 and critical variants	for $14$0, $14$1 and $14$2 for every $14$3

A unifying feature is the proper-coloring constraint, but the optimization target varies sharply: minimum palette size, minimum number of colors used online, minimum cumulative score, perpetual survivability under recoloring, or convergence time under decentralized updates. Several sources state explicitly that structure dominates order: in the token model, the number of vertices matters only indirectly through forbidden induced subgraphs; in the game-vertex-criticality setting, “structure (not order) drives the game parameters” (Milans et al., 2017, Jakovac et al., 2021).

2. Painter–Builder proper coloring on 14 vertices

In the unbiased Painter–Builder $14$4-coloring game, the board is the empty graph on the fixed vertex set $14$5. Rounds are alternating, with Painter moving first. In each round Painter must color exactly one previously uncolored vertex with one of $14$6 colors, and then Builder adds exactly one edge between two previously unconnected vertices. At every moment the partial coloring must be proper, recoloring is not allowed, and the game ends when either all $14$7 vertices are colored or Painter has no legal move. Equivalently, Painter loses if and only if there exists an uncolored vertex $14$8 whose currently colored neighbors collectively use all $14$9 colors (Bednarska-Bzdęga et al., 2016).

For general $f(w,G)=w$0, the paper proves that $f(w,G)=w$1 is of logarithmic order and, for every $f(w,G)=w$2,

$f(w,G)=w$3

For the concrete case $f(w,G)=w$4, the asymptotic lower bound is vacuous, but the paper’s explicit specialization gives

$f(w,G)=w$5

and therefore $f(w,G)=w$6 colors suffice for Painter against worst-case Builder. A second lower bound, obtained by specializing the biased theorem to $f(w,G)=w$7, yields

$f(w,G)=w$8

so $f(w,G)=w$9. The resulting rigorous interval is

$s(P_{14})=21$0

The upper bound is witnessed by the greedy-random “smallest available color” strategy. Painter colors the first vertex with color $s(P_{14})=21$1. When Builder adds an edge $s(P_{14})=21$2, Painter chooses one endpoint uniformly at random if both are uncolored, the unique uncolored endpoint if exactly one is uncolored, and otherwise any uncolored vertex; she then assigns the smallest available color. For $s(P_{14})=21$3, the bad-event analysis gives

$s(P_{14})=21$4

Hence Builder cannot have a sure-win strategy, implying that Painter has a winning strategy with $s(P_{14})=21$5 colors.

The lower-bound machinery is asymptotic and does not settle the small instance tightly. The waiting-room and color-neighborhood escalation lemmas require large sets such as $s(P_{14})=21$6 and constants like $s(P_{14})=21$7; the source states that for $s(P_{14})=21$8 these preconditions fail. Accordingly, the paper does not determine whether $s(P_{14})=21$9 suffices, and it explicitly notes that the constants in Theorem 1 were not optimized. In biased variants, however, the behavior is clearer: in the $s(K_{14})=105$0 game, two colors suffice for Painter, while in the $s(K_{14})=105$1 game the theorem gives

$s(K_{14})=105$2

3. Online token graph coloring on 14-vertex host graphs

The online token graph model fixes a host graph $s(K_{14})=105$3 and proceeds by alternating Spoiler and Algorithm moves. Spoiler places a token on a vertex of $s(K_{14})=105$4, and Algorithm immediately colors that token. Tokens on the same vertex, or on adjacent vertices in $s(K_{14})=105$5, must receive distinct colors. The token graph $s(K_{14})=105$6 has the played tokens as vertices, with adjacency defined by host-graph distance at most $s(K_{14})=105$7; Spoiler must maintain $s(K_{14})=105$8 throughout play. The value $s(K_{14})=105$9 is the minimum number of colors that an optimal Algorithm can guarantee against worst-case Spoiler (Milans et al., 2017).

A graph $\chi_g^{\infty}(P_{14})=\chi_g^{\infty}(C_{14})=4$0 is online-perfect if $\chi_g^{\infty}(P_{14})=\chi_g^{\infty}(C_{14})=4$1 for every $\chi_g^{\infty}(P_{14})=\chi_g^{\infty}(C_{14})=4$2. The structural characterization in the paper states that this is equivalent to forbidding induced subgraphs isomorphic to odd cycles $\chi_g^{\infty}(P_{14})=\chi_g^{\infty}(C_{14})=4$3 for odd $\chi_g^{\infty}(P_{14})=\chi_g^{\infty}(C_{14})=4$4, $\chi_g^{\infty}(P_{14})=\chi_g^{\infty}(C_{14})=4$5, $\chi_g^{\infty}(P_{14})=\chi_g^{\infty}(C_{14})=4$6, and the bull graph $\chi_g^{\infty}(P_{14})=\chi_g^{\infty}(C_{14})=4$7, and also equivalent to $\chi_g^{\infty}(P_{14})=\chi_g^{\infty}(C_{14})=4$8 being obtainable from a bipartite graph by cloning vertices. This yields a direct decision procedure for $\chi_g^{\infty}(P_{14})=\chi_g^{\infty}(C_{14})=4$9-vertex host graphs. If $14$0 is bipartite, then $14$1 is online-perfect and $14$2 for all $14$3; the source lists $14$4 and $14$5 as explicit examples. If $14$6 contains an induced $14$7, then

$14$8

and if $14$9, then $G$0. If $G$1 contains an induced $G$2, then for even $G$3,

$G$4

The paper emphasizes that the parameter $G$5 and the host-graph structure dominate the role of the vertex count. For a given $G$6-vertex graph, the recommended checklist is: check bipartiteness; search for the forbidden induced subgraphs; if none occur, conclude that $G$7; if some obstruction occurs, use the obstruction-specific lower bounds and, where needed, the fractional coloring upper bound

$G$8

The examples in the source make the point sharply: $G$9 and $14$00 are online-perfect, a $14$01-vertex graph containing an induced $14$02 is not online-perfect, and a $14$03-vertex chordal graph may still fail to be online-perfect because the bull graph is chordal and forbidden.

4. Slow-coloring and score-based 14-vertex instances

In the slow-coloring game, Lister and Painter play on a simple graph $14$04. In each round, Lister marks a nonempty set $14$05 of currently uncolored vertices and scores $14$06 points; Painter then colors any independent subset $14$07. Colored vertices are deleted, and the game ends once all vertices are colored. The value under optimal play is the sum-color cost $14$08 (Mahoney et al., 2015).

The paper supplies both general inequalities and exact $14$09-vertex evaluations. With $14$10, independence number $14$11, and Hall ratio

$14$12

the general bounds are

$14$13

For $14$14-vertex examples, the paper records exact values or tight intervals:

$14$15

For the even cycle $14$16, the source gives

$14$17

For complete bipartite graphs with $14$18 and $14$19,

$14$20

Two worked examples are

$14$21

The paper also gives a practical recipe for arbitrary $14$22-vertex graphs: compute $14$23 to obtain the universal lower bounds

$14$24

then compute $14$25 for the upper bound $14$26. For trees on $14$27 vertices, the extremal theorem specializes to

$14$28

This score-based model differs from palette-minimization games, but it again shows that a fixed order does not determine the outcome: the range from $14$29 for the star to $14$30 for the clique is entirely structural.

5. Eternal recoloring and distributed network dynamics

The eternal vertex coloring game introduces repeated rounds. Fix $14$31 colors. In every round, each vertex is chosen exactly once; if a vertex is uncolored it must be colored properly, and if it is already colored with $14$32 it must be recolored to a color different from $14$33 and different from the colors on its neighbors. Alice wants to maintain a proper coloring forever, and Bob wants to force failure. The minimum such $14$34 in the standard A-game is the eternal game chromatic number $14$35 (Klostermeyer et al., 2018).

For $14$36-vertex graphs, the paper gives several exact values. Because $14$37 is even, Alice starts every round in the A-game. The listed evaluations are

$14$38

For $14$39, the paper provides

$14$40

The general upper bound is

$14$41

while Theorem 3.10 states that if $14$42 is connected and not $14$43, $14$44, or $14$45, then $14$46. The source also records large separations between one-shot and eternal play, for example $14$47 classically but $14$48 grows with $14$49 according to parity formulas.

A different recurrent model is the network coloring game, where all vertices act simultaneously in synchronous rounds, each vertex sees only the colors chosen by its neighbors, and a vertex is happy when no neighbor shares its color. The note studies the Frugal strategy: if a vertex is happy it sticks, and if it is unhappy it samples uniformly from

$14$50

With $14$51, the convergence time $14$52 to a proper coloring satisfies $14$53 with high probability, and for any $14$54,

$14$55

where $14$56 (Fryganiotis et al., 2021).

Specialized to $14$57, the note recommends setting $14$58. It then states

$14$59

as a worst-case analytic guarantee, while emphasizing that this constant is “very loose.” The same theorem yields $14$60 rounds with high probability for arbitrary $14$61-vertex graphs, including low-degree cases such as $14$62 and $14$63 and the extreme case $14$64 with $14$65 and $14$66.

6. Standard game chromatic number, bicolored-subgraph methods, and 14-vertex criticality

In the standard graph coloring game, Alice and Bob alternately color previously uncolored vertices from a fixed palette of $14$67 colors, always maintaining a proper coloring. Alice wins if all vertices are colored; Bob wins if some uncolored vertex has all $14$68 colors present in its neighborhood. The associated invariant is the game chromatic number $14$69, with the basic bounds

$14$70

A major upper-bound method uses a proper $14$71-coloring $14$72 of $14$73 and the bicolored subgraphs $14$74. If every bicolored subgraph $14$75 satisfies $14$76 via reactive strategies, then

$14$77

and without reactivity,

$14$78

For acyclic colorings, this recovers

$14$79

The paper gives concrete $14$80-vertex implications: an acyclic $14$81-coloring yields $14$82, an acyclic $14$83-coloring yields $14$84, and if every bicolored subgraph is a matching then $14$85 (Bradshaw, 2020).

Game-vertex-criticality studies how these invariants change under vertex deletion. For $14$86, a graph is $14$87-$14$88-game-vertex-critical if $14$89 for every vertex $14$90. The paper shows that for $14$91 the difference $14$92 can be arbitrarily large, and it provides exact characterizations of all $14$93-$14$94-game-vertex-critical graphs and connected $14$95-$14$96-lower-game-vertex-critical graphs (Jakovac et al., 2021).

At order $14$97, the canonical example is the crown graph $14$98, where $14$99 is a perfect matching. The source states:

$2 \le k_{\min}(14)\le 5$00

and also

$2 \le k_{\min}(14)\le 5$01

with

$2 \le k_{\min}(14)\le 5$02

Thus the same $2 \le k_{\min}(14)\le 5$03-vertex graph is simultaneously a $2 \le k_{\min}(14)\le 5$04-$2 \le k_{\min}(14)\le 5$05-lower-game-vertex-critical graph and a $2 \le k_{\min}(14)\le 5$06-$2 \le k_{\min}(14)\le 5$07-lower-game-vertex-critical graph. By contrast, the paper’s small-$2 \le k_{\min}(14)\le 5$08 characterizations exclude many possible $2 \le k_{\min}(14)\le 5$09-vertex critical examples: the only $2 \le k_{\min}(14)\le 5$10-$2 \le k_{\min}(14)\le 5$11-game-vertex-critical graph is $2 \le k_{\min}(14)\le 5$12 for all four invariants considered, and there is no connected $2 \le k_{\min}(14)\le 5$13-vertex $2 \le k_{\min}(14)\le 5$14-lower-game-vertex-critical graph for $2 \le k_{\min}(14)\le 5$15, $2 \le k_{\min}(14)\le 5$16, $2 \le k_{\min}(14)\le 5$17, or $2 \le k_{\min}(14)\le 5$18 because the classified families have smaller orders.

Across these models, the most stable conclusion is that “14-vertex graph coloring game” is not a single invariant-bearing object but a family of structurally sensitive finite games. The recurring theme is the same in each framework: order $2 \le k_{\min}(14)\le 5$19 is large enough to exhibit nontrivial extremal behavior, but exact outcomes are driven by forbidden induced subgraphs, bipartite structure, degree, treewidth of bicolored layers, or specific constructions such as the crown graph.